COLLEGE     ALG-EBEA-. 


BY 


LEONARD  EUGENE  DICKSON,   Ph.D., 

ASSISTANT   PROFESSOR  OF  MATHEMATICS  IN 
THE  UNIVERSITY  OF  CHICAGO. 


SECOND  EDITION,   CORRECTED. 


NEW   YORK: 
.       JOHN   WILEY   &   SONS. 
London  :   CHAPMAN  &  HALL,  Limited. 


X^iS 


Copyright,  1902, 

BY 

L.  E.  DICKSON. 


PRESS  OP 

BRAUN WORTH  h  CO. 

BOOK  MAMUFACTURERB 

BflOOKLVN,  N.  V. 


PREFACE. 


This  text  is  intended  primarily  for  the  college  and  the  tech- 
nical school.  By  treating  only  the  subjects  usually  given  in  the 
college  course  in  algebra,  space  has  been  gained  for  a  more  detailed 
exposition  of  the  more  difficult  topics.  As  the  extent  and  the 
character  of  the  review  at  the  beginning  of  the  course  upon  topics 
prescribed  for  entrance  to  college  varies  so  widely,  and  as  the 
review  is  usually  conducted  in  an  informal  manner,  it  seemed  best 
to  the  author  to  leave  to  the  instructor  the  review  of  the  ele- 
mentary principles,  but  to  give  in  the  text  review  exercises. 

x\s  to  the  order  of  the  subjects,  the  aim  has  been  to  present 
first  those  topics  which  are  readily  mastered  by  the  student,  and  to 
reserve  for  the  latter  part  of  the  text  the  questions  grouped  around 
the  subjects  involving  infinite  series.  In  reviewing  the  subject 
of  simultaneous  equations,  the  student  is  led  naturally,  almost 
unavoidably,  to  the  determinant  notation.  Determinants  are  there- 
fore treated  early  in  the  text,  an  order  of  presentation  shown  by 
actual  experience  to  give  very  satisfactory  results,  especially  in 
arousing  the  interest  of  the  student. 

In  the  chapter  on  graphic  algebra,  the  first  principles  of 
coordinate  geometry  are  introduced  and  applied  to  the  study  of 
simultaneous  equations  and  inequalities.  In  this  connection  is 
presented  an  elementary  account  of  the  solution  of  numerical  equa- 
tions, chiefly  from  the  graphic  standpoint.  The  arrangement  is 
such  as  to  admit  of  a  very  brief  course  or  of  a  fuller  course 
involving  Horner's  method  of  synthetic  division.     The  practice  of 


474!l^bo 


IV  PREFACE, 

emphasizing  graphic  algebra  in  courses  for  technical  students  com- 
mends itself  also  to  the  general  student.    • 

Attention  is  invited  to  the  proofs  of  the  fundamental  theorems 
on  logarithms,  the  treatment  of  mathematical  induction  and  the 
illustrations  showing  the  necessity  of  the  different  steps  in  the 
process,  the  examples  from  the  physical  sciences  of  the  topic  varia- 
tion, the  complete  proof  of  the  general  binomial  theorem  independ- 
ent of  the  principle  '^permanence  of  form,"  the  establishment  of 
the  relations  between  the  roots  and  the  coefficients  of  the  quad- 
ratic, cubic,  and  quartic  equations  prior  to  the  proof  of  the  general 
theorem,  the  solution  of  those  equations  before  introducing  the 
assumption  that  every  equation  has  a  root  (here  proved  in  the 
Appendix). 

The  attempt  has  been  made  to  present  the  subjects  limits  and 
infinite  series  in  as  simple  form  as  is  consistent  with  rigor. 

Forty-five  sets  of  exercises,  averaging  over  fifteen  to  a  set,  are 
given  at  very  short  intervals  in  the  text.  Some  historical  data 
have  been  introduced,  with  no  attempt  to  give  the  source,  the 
subject-matter  being  classical. 

The  author  is  under  great  obligation  to  Dr.  Moulton,  of  the 
Department  of  Astronomy  of  the  University  of  Chicago,  who  read 
with  care  the  entire  manuscript  and  offered  numerous  suggestions 
as  to  the  form  of  presentation,  most  of  which  have  been  adopted. 
Likewise  the  thanks  of  the  author  are  due  Professor  J.  W.  A. 
Young  of  the  same  University,  who  examined  the  more  critical 
chapters  and  offered  valuable  suggestions. 

Proof-sheets  of  the  entire  book  were  carefully  read  by  Professor 
L.  L.  Conant  of  the  Worcester  Polytechnic  Institute,  whose  correc- 
tions and  suggestions  have  led  to  numerous  improvements.  Finally, 
the  author  is  indebted  to  Professors  Moore  and  Young  of  the  Uni- 
versity of  Chicago,  who  examined  critically  the  proof-sheets  of 
certain  portions  of  the  text. 

CuiCAGO,  Januarv,  1902. 


CONTENTS. 

[See  the  Index  at  the  end  of  the  book.] 

PAGE 

Kumber  in  Algebra;  Surds  and  Imaginaries 1 

Exponents;  Logarithms 10,  17 

Table  of  Four- place  Logarithms 24,  25 

-^  ilL  Factor  Theorem;  Quadratic  Equations 27,  29 

IV.  Simultaneous  Equations;  Determinants 35 

V.  Ratio;  Proportion;  Variation 58-61 

VI.  Arithmetical,  Geometrical,  and  Harmonical  Progressions.. . : ..  64-71 

VII.  Compound  Interest  and  Annuities. 73 

VIII.  UndeterminM  Coefficients;  Partial  Fractions 76,  80 

IX,  Permutations  and  Combinations 85 

Binomial  Theorem  for  Positive  Integral  Index 90 

Multinomial  Theorem 92 

X.  Probability  (Chance) 94 

***   XI.  Mathematical  Inductitm 99 

■*■  XII.  Limits;  Indeterminate  Forms 104,  106 

XIII.  Convergency  and  Divergency  of  Series 113 

XIV.  Power  Series;  Undetermined  Coefficients 125,  126 

Expansion  into  Series;  Reversion  of  Series 126,  128 

XV.  Binomial  Theorem  for  Any  Index 130 

XVI.  Exponential  and  Logarithmic  Series 136,  140 

Natural  Logarithms;  Interpolation 141,  142 

XVII    Summation  of  Series;   Recurring  Series;   Generating  Relation; 

Generating  Fraction 145,  148 

The  Method  of  Differences 148 

XVIII.  Graphic  Algebra. 

Coordinates;  Graph;  the  Straight  Line 152-160 

Simultaneous  Equations 164 

Simultaneous  Inequalities 166 

Solution  of  Numerical  Equations 168 

Horner's  Method  of  Synthetic  Division 169 

Descartes'  Rule  of  Signs;  Location  of  Roots 176 

V 


vi  CONTENTS. 

CHAPTSB  PAGE 

XIX.  Theory  of  Equations. 

Solution  of  Cubic  and  Quartic  Equations 180,  185 

Solution  of  Certain  Equations  of  the  Fifth  Degree 189 

Reciprocal  Equations 190 

Fundamental  Theorem  of  Algebra 193 

Relations  between  the  Roots  and  the  Coefficients 194 

Fractional,  Surd,  and  Imaginary  Roots 198,  199 

Symmetric  Functions  of  the  Roots 200 

Miscellaneous  Exercises 205 

APPENDIX. 

Argand*s  Diagram 207 

De  Moivre's  Theorem 208 

Solution  of  Cubic  in  the  Irreducible  Case « .  .•  210 

Proof  that  Every  Equation  has  a  Root 211 

Index ••• 213 


SYMBOLS  AND  ABBREVIATIONS. 


=  ,  equal.  JJr^  nPn  S6. 

^,  not  equal.  A.  P.,  G.R,  H.P.,  64. 

iz  ,  approaches,  106.  sin,  cos,  208. 

00  ,  infinity,  69,  105.  1^1  —  absolute  value  of  t,  114. 
6-2.71828.,..,  137.  w!  =  1  X  2  x  3  X  . .  .X  ^.,  80. 

1  —  |/  — 1. 


I 


COLLEGE  ALGEBRA. 


CHAPTER   I. 

NUMBER  IN  ALGEBRA;    SURDS   AND   IMAGINARIES. 

1.  The  natural  numbers  or  positive  integers  (1,  2,  3, ...)  make 
it  possible  to  enumerate  the  objects  of  a  group  considered  for  the 
time  as  equivalent  entities.  It  has  been  established  that  primitive 
counting  was  done  on  the  fingers  and  that  in  many  languages  the 
numeral  ">  is  merely  the  word  for  hand,  10  for  both  hands,  and  20 
for  the  whole  man  (hands  and  feet).*  AVhile  5,  10,  20,  and  60  have 
been  used  as  bases,  10  is  the  usual. base. 

If  a  group  of  objects  can  be  partitioned  into  equivalent  smaller 
groups,  each  smaller  group  or  a  combination  of  them  is  a  fraction 
(I,  f,  1^,  ...)  of  the  original  group.  Abstractly,  a  fraction  is  the 
quotient  of  two  positive  integers.  Fractional  results  may  or  may 
not  admit  of  an  interpretation  in  a  particular  problem.  A  shepherd 
would  declare  it  to  be  impossible  to  separate  his  flock  of  50  sheep 
into  three  equal  flocks  ;  but  would  find  no  theoretical  difficulty  in 
dividing  a  50-foot  rope  into  three  equal  pieces.  The  algebraic 
statement   for  each  problem   is   the  same  :    to   find  x  such  that 

*Thus  in  the  language  of  the  Tamanacs,  the  word  for  6  is  "one  on  the 
other  hand  ";  the  word  for  12  is  ''two  to  the  foot  ";  for  15,  **a  whole  foot  "; 
for  16,  ''one  to  the  other  foot  ";  the  word  for  20  is  "one  Indian";  for  40, 
"two  Indians  ";  etc. 


2  NUMBHt^'  fN  '^bpiBkA:  ^-SJURD^-  AND  IM  AGIN  ARIES,        [Ch.  I 

3a;  =  50.     The  formal  solution  is  :?:  —  ^-  ;    the  possibility  of  its 

interpretation  depends  upon  the  character  of  the  special  problem. 

The  Egyptians  used  fractious  before   1700  B.C.*  and  resolved 

them  into  sums  of  unit  fractions.     Thus  f  was  written  3  15,  which 

meant  \  +  -^^.     By  the  Babylonians,  f  was  written  37  30,  which 

37       30 
meant  -^  -f  ^^^     Instead  of  fractions,  the  Greek  geometers  used 

the  ratio  of  two  numbers  or  two  magnitudes. 

For  the  introduction  of  negative  numbers  (as  well  as  the  decimal 
positional  system  and  the  symbol  0),  algebra  is  indebted  to  the  early 
mathematicians  of  India  (between  500  and  600  a.d.).  We  now' 
find  it  convenient  to  write  —  15°  for  15°  below  zero  ;  —  100  ft.  for 
100  feet  below  sea-level,  thereby  abbreviating  our  map  notations. 
The  determination  of  a  number  x  such  that  h  -\- x  =  c  leads  to  the 
algebraic  solution  x  =  c  —  h.  li  h  exceeds  c,  the  result  c  —  h  \^  a 
negative  number.  Such  a  result  would  be  excluded  if  the  problem 
were  to  find  how  many  feet  of  rope  must  be  added  to  a  rope  h  feet 
long  to  make  a  rope  c  feet  long.  But  the  negative  result  leads  us 
to  restate  the  problem  so  that  the  required  c-ioot  rope  is  seen  to  be 
obtained  by  cutting  o^h  —  c  feet  of  rope  from  the  Z>-foot  rope. 

In  this  connection,  we  note  the  Eoman  notations  IV  for  4,  VI 
for  6,  IX  for  9,  XI  for  11,  which  seem  to  have  been  of  Etruscan 
origin. 

The  term  rational  number  includes  positive  and  negative  in- 
tegers and  fractions.     All  other  numbers  are  called  irrational. 

The  solution  of  equations  of  the  form  x''' —  A^  where  A  is  a 
rational  number  and  n  a  positive  integer,  introduces  two  classes  of 
irrational  numbers.  Thus,  for  ^^  =  2,  and  A  a  positive  integer  not 
the  square  of  a  rational  number,  the  square  root  of  A^  denoted  by 
the  symbol  X^A,  is  an  irrational  number  called  a  quadratic  surd. 
Similarly,  4'^2,  V  —  ^,  VA,  A  not  the  cube  of  a  rational  number, 


*Rhind  papyrus,   *' Directions  for  Attaining  to  the   Knowledge  of  All 
Dark  Things." 


Sec.  2]  COLLEGE  ALGEBRA.  3 

are  surds  of  the  third  order.  In  general,  V A,  where  A  is  positive 
if  n  is  even,  is  a  surd  of  order  n.     The  second  class  of  irrational 

n  . ■ 

numbers  are  defined  by  the  symbols  V A,  where  A  is  negative  and 
n  even  (§  4). 

While  the  equation  x^  =  80  possesses  the  formal  solutions 
x=±  1^80  (the  positive  root  is  designated  VSO,  the  negative  —  VSO), 
the  possibility  of  the  interpretation  of  one  or  both  results  depends 
upon  the  character  of  the  particular  problem.  It  is  possible  to  form 
a  square  of  area  80  square  feet,  but  impossible  to  arrange  80  square 
blocks  of  equal  size  in  the  form  of  a  square  and  yet  preserve  the 
form  of  each  block. 

2.  Tlie  fact  that  surds  really  exist  as  such  may  be  illustrated  by 
showing  that  V2  is  not  expressible  as  a  rational  number.  If  we 
take  the  side  of  a  square  as  the  unit  of  length,  the  diagonal  is  of 
length  V2,  But  it  is  proved  in  Geometry  that  the  side  and  diago- 
nal are  incommensurable  (see  §  55).  Hence  1  and  l/2  have  no 
common  measure.  It  follows  that  V2  is  not  equal  to  a  rational 
number.     For,  if 

(1)  ^-^  =  1^ 

then  —  would  be  contained  p  times  in  V2  and  q  times  in  1  and  hence 

be  a  common  measure  of  1  and  V2, 

To  give  a  purely  algebraic  proof,  suppose  that  equation  (1)  holds 

p     .        .     . 
true,  the  fraction  —  being  in  its  lowest  terms,  so  that  p  and  q  are 

integers  having  no  common  divisor.  By  squaring  and  multiplying 
by  q'^,  we  get  2q^  =  p^,  so  that  7/  and  therefore  p  is  divisible  by  2. 
Setting  p  =  2r,  we  get  q^  —  2r^,  so  that  q  is  divisible  by  2.  Then 
p  and  q  have  a  common  divisor  2,  contrary  to  hypothesis. 

3.  But  with  the  introduction  of  rational  numbers  and  surds,  we 
do  not  meet  all  the  demands  which  are  made  upon  a  number  system. 
rWe  are  led  in  Geometry  to  consider  the  number  n  which  expresses 


4  NUMBER  IN  ALGEBRA;    SURDS  AND  IM AGIN  ARIES,  ^     [Ch.  I 

the  ratio  of  the  circumference  to  the  diameter  of  a  circle  and  to 
approximate  its  vahie  by  considering  the  perimeters  p^  and  P^  of  an 
inscribed  and  a  circumscribed  regular  hexagon,  the  perimeters  jOj, 
and  Pj2  ^^  ^^  inscribed  and  a  circumscribed  regular  polygon  of  12 
sides,  etc.  From  the  results,  true  to  four  decimal  places,  for  a 
circle  of  unit  diameter: 

A  =  3  ^    Pu  =  3.1058,   .  .  .  ,    p,,,  =  3.1415,    .  .  . 

Pg  =  3.4641,   F,,  =  3.2153,   .  .  .  ,   F,,,  =  3.1416,   .  .  . 

we  obtain  a  succession  of  numbers  between  each  pair  of  which  the 
value  of  7t  must  lie.     By  proving  that 

P,<   Pu<   P2,<    '    -    <    P^M  <   .    .    .    <  ^, 
P,  >   P,,  >  P,,   >    .    .    .    >   P33,  >    .    ,,>7t, 

and  that  the  diiference  P„  —  Pn  can  be  made  to  differ  from  zero  a? 
little  as  we  please  by  sufficiently  increasing  the  number  of  sides  w, 
we  have  pointed  out  to  us,  with  as  great  a  degree  of  precision  as  we 
may  desire,  a  certain  limit,  which  we  take  as  the  value  of  n. 

In  an  analogous  manner,  we  can  define  the  number  1^2  by  means 
of  two  sequences  of  rational  numbers. 


1, 

1.4, 

1.41, 

1.414, 

1.4142, 

1.41421, 

3, 

1.5, 

1.42, 

1.415, 

1.4143, 

1.41422, 

By  the  arithmetical  process  for  the  extraction  of  a  square  root,  wo 
find  that  the  value  of  1^2  lies  between  each  pair  of  corresponding 
numbers  in  the  sequences. 

In  general,  two  such  sequences  of  rational  numbers  proceeding 
by  a  given  law  are  said  to  define,  by  a  limiting  process,  a  number.* 
The  value  of  the  number  may  be  determined  to  as  great  a  degree  of 
approximation  as  may  be  desired.  All  such  numbers  as  well  as  all 
rational  numbers  are  called  real  numbers. 

♦The  above  sequences  which  defined  the  number  n  can  evidently  be 
replaced  by  sequences  of  rational  numbers  related  to  the  Pn  and  P„ . 


Sec.  4]  COLLEGE  ALGEBRA,  5 

4.  An  even  root  of  a  negative  number  is  called  an  imaginary 

4  .     6   

quantity.  Thus  y  —  1,  r  —  2,  r  —  1  are  imaginaries.  Unlike 
surds  and  other  real  numbers,  an  imaginary  can  not  be  expressed 
approximately  in  terms  of  rational  numbers  and  hence  has  no  inter 
pretation  in  strictly  arithmetical  problems.  By  the  introduction 
of  imaginaries,  we  may  give  a  formal  solution  of  the  equations 
a:*  =  ~  1,  a;'^  :=  —  2,  and,  indeed,  of  every  quadratic  equation. 
By  extending  the  system  of  all  real  numbers  by  the  introduction  of 
the  quantity  1^  —  1,  we  obtain  the  quantities  a  -{-I  V  —  I,  where 
a  and  h  are  arbitrary  real  numbers.  We  shall  see  that  the  system 
of  these  complex  quantities  a  -{-  h  V  —  1  forms  a  number  system 
within  which  may  be  performed  all  algebraic  operations  including 
the  solution  of  all  algebraic  equations,  so  that  a  further  extension 
is  unnecessary.  The  employment  in  algebra  of  imaginaries  has 
therefore  a  great  practical  value  in  that  the  operations  may  be 
effected  without  the  limitations  otherwise  necessary.  To  further 
justify  this  extension,  we  recall  that  negative,  fractional,  and  surd 
numbers  were  introduced  to  enable  us  to  give  a  formal  solution  of 
many  simple  problems  which  would  otherwise  have  remained 
insolvable,  and  that  the  possibility  of  the  interpretation  of  nega- 
tive, fractional,  or  irrational  results  depends  upon  the  nature  of 
the  particular  problem.* 

5.  If  a  -\-  Vb  =  c  -{-  Vd,  where  a,  h,  c,  d  are  rational  numbers 
and  Vb  is  irrational^  then  a  =  c,  b  =  d. 

From  a  —  c  -\-  Vb  =  Vd,  we  derive,  after  squaring, 
2(a  -  c)  Vb  =  d-b-{a-  cy. 

Unless  the  coefficient  a  —  c  is  zero,  we  could  express  Vb  as  a  rational 
number,  contrary  to  assumption.     Hence  a  =  c,  so  that  b  =  d. 

*  A  possible  interpretation  of  complex  qaantities  is  given  in  the  Appendix. 
The  instructor  may  prefer  the  illustration  by  means  of  operations  which  com- 
bine a  rotation  with  a  magnification.  Thus  —  1  rotates  through  180°,  V  —  1 
through  90°,  4  -}-  3  V  —  i  magnifies  five-fold  and  rotates.  See  Chrystal's 
Algebra,  I,  p.  239. 


6  NUMBER  IN  ALGEBRA;    SURDS  AND  IMAClNARIBS,      [Ch.  I 

6.  Let  us  attempt  to  extract  the  square  root  of  a  -{-  Vb^  where 
Vb  is  a  true  surd  and  a  is  rational.  We  seek  a  result  of  the  form 
Vet  -j-  V fd  in  whieli  a  and  /3  are  rational.     Setting 


i/a  +  Vb  :=  Va  +  V/3, 
and  squaring,  we  find  that 

By  the  above  theorem,  we  may  equate  the  rational  parts  and  also 
the  irrational  (surd)  parts.     Hence 

a  =  a  -\-  ft^         b  —  4:a/3. 
Then  {a  -  f3f  =  (tY  +  I3f  -  4:a/3  =  a^^  -  b,  so  that 


az=^{a  +  Va^  -  b),         13  =  ^{a  -  Vd^  -  b). 

By  assumption  a  and  jS  are  rational.  Hence  the  square  root  of 
a  -{-  Vb  is  expressible  as  a  sum  of  tivo  quadratic  surds  Voc  -\-  V ft 
if,  and  only  if,  c?  —  b  is  the  square  of  a  rational  number. 

For  example,  if  aj  ==  6,  Z>  =  20,  c?  —  b  is  the  square  of  4. 
Hence  a  +  |/^  =  6  +  l/20  is  the  square  of  V'oc  -\-  V~ft  =  Vh  -\-  \. 

When  the  problem  is  solvable,  it  may  usually  be  done  by  inspec- 
tion as  follows.  Put  the  expression  a-\-  Vb  into  the  form  m  -\-  2  Vn 
by  taking  m  =  a,  n  =  ^/4.  The  required  root  is  Vol  -f  V ft,  where 
a  -\-  ft  ^=^  m,  aft  =  /^. 

Thus  6  +  ^^20  =  6  +  2  |/5  =  (1  +  Vlf, 

since  1  -}-  -5  =  6,  1  •  5  —  5. 

Tims  16  +  6  V7  =  16  +  2  V'^  =^{Vl  +  V'^y\ 
sine:)  7  +  9  =  16,  7- 9  =r  63. 

7.  Denote  by  i  the  symbol  V —1.  Then  +  i  and  —  i  are  the  roota 
of  :c^  r=  —  1.  By  the  symbol  V  —  c^  where  c  is  positive,  we  shall 
mean  V  —  I  Vc=  i  Vc,  where  Vc  denotes  the  positive  square  root 
of  c.     Thus 

i2=~l,   i^=-i,   i^  =  +  l,   22»=  (-!)«,    ^2^  +  ^  =  ^(-l)^ 


Sec.  8]  COLLEGE  ALGEBRA,  7 

If  c  and  d  are  any  two  positive  real  quantities, 

V~^  V  —  d  —  i  Vc  'iVd=:  —  V'cVd-=  --  V7d. 
In  introducing  the  equation  x^  =^  --  c  as  an  equation  to  be  solved 
by  algebra,  we  are  tacitly  assuming  that  x  may  be  combined  with  itself 
and  with  real  numbers  according  to  the  laws  of  algebra  for  the  com- 
bination of  real  numbers,  so  that  d  V  —  c—  V  —  c  d^  d -\-  V  —  c  = 
V  —  c  -\-  d^  etc.  In  addition  to  these  assumptions,  we  assume  that 
complex  quantities  may  be  combined  according  to  the  laws  holding 
for  real  numbers.     Then 

{a  +  bi)  ±  (^  +  fti)  ^{a±a)-^{h±  ft)U 
{a  +  hi)  {a  +  I3i)  =  {aa  -  1(5)  +  (aft  +  la)i, 
a  +  [51  _  {a  -\-l3i){a  —  hi)  _aa  -\-h/3      aft  —  ha. 
a  +  hi  ~~  {a  +  hi)  {a  -  hi)  ~  d^  +  U^  "*"   d^  +  W  ** 
Hence  the  sum,  differe?ice,  product,  or  quotient  of  any  ttoo  comj^lex 
quantities  is  itself  a  comi)lex  quantity.'^ 

In  freeing  the  denominator  of  the  above  fraction  from  imagi- 
naries,  we  used  the  multiplier  a  —  hi,  called, the  conjugate  of  the 
denominator  a  -f  hi.  The  sum  and  the  product  of  two  conjugate 
complex  quantities  are  both  real. 

8.  TJie  three  cuhe  roots  of  unity  are 

1,  G9  =  -  i  +  i  V"^=~3,      G?'^  =  -  I  ~  1  l/^Ts, 

so  that  G3^  —  \,       1  +  &?  -|-  G9^  =  0. 

The  roots  of  x^  =  1  are  :?:  =  1  and  the'  two  roots  of 

'^ -,  =  ^2  _[_  ^  _j_  1  ^  0. 

X  —  1 

Completing  the  square  in  the  quadratic  equation,  we  get 

Hence  x=— ^±V  —  ^.      Setting  -  ^  +  ^  V  ~  3  -  oa,   the 
second  root  is 

- 1  - 1  V'^^3  =  ( _  ^  + 1  i/Tr3)2  ^  ^, 

9.  I7i  an  equation  hetiueen  two  comjjlex  quantities,  the  real  parts 
are  equal  and  also  the  imaginary  parts. 

*  The  complex  quantity  a  +  hi  is  real  if  &  =  0. 


8  NUMBER  IN  ALGEBRA;    SURDS  AND  IMAGINARIES.        [Ch.  I 

Let  a-^-  bi  =  a-}-  ^i,  where  a,  b,  a,  /3  are  real  numbers.     Then 
a  —  a  z=  (/3  —.b)i. 
Upon  squaring,  we  find  that  the  number  {a  —  a)^,  which  is  positive 
or  zero,  must  equal  the  number  —  (/3  —  by^  which  is  negative  or 
zero,  so  that  each  must  be  zero.     Hence  a  =  or,  b  ^  ft. 

In  particular,  \i  a -\-  bi  —  0,  then  a  =  Q,  b  =  0, 

10.  The  square  root  of  any  complex  quantity  may  ahvays  be  ex- 
pressed as  a  complex  quantity.^ 

Let  the  given  complex  quantity  be  «  +  bi,  where  a  and  b  are 
real  and  i  =  V  —  l»  We  seek  real  numbers  o^  and  y  which  will 
make 


Va  -j-  bi  =  X  -\-  yi. 
Squaring,  a  -\- bi  =  x^  —  y^  -{-  2xyi, 

Equating  the  real  parts  and  also  the  imaginary  parts, 

^2  _  y2  —  f^^     2xy  =  b. 
Then  {x^  -  y^  +  ^.x^  =  {x^  +  y^f  =  a'^  +  P. 

Since  x  and  y  are  to  be  real,  x^  +  y^  must  be  positive.     Hence 

a;2  -|-  ^2  _  |/^2  _j_  ^2  (positive  square  root). 

Having  the  sum  and  difference  of  x^  and  y^,  we  derive 

„        Va^  +  b^^  +  a        ,        Va^~+~^  -  a 
x^  = 2 >     f  = -2 • 


Since  Va^  +  b^  is  positive  and  greater  than  a,  the  expressions  for  x^ 
and  y^  are  positive,  so  that  real  values  of  x  and  y  may  be  determined 
by  extracting  the  square  roots  of  positive  quantities.    Since  2xy  =  Z>, 
the  sign  of  y  is  determined  as  soon  as  the  sign  of  x  is  chosen. 
Hence  there  are  always  two  and  only  two  square  roots  of  a  -\-  bi. 
For  example,  to  find  the  square  roots  of  5  —  12i,  we  have 
x^  —  y^  =  5,     2xy  =  —  12, 
whence  x^  -}-  y^  =  13,    x^  =  9,    y^  =  4.      The   square   roots   are 
±  (3  -  2i), 

*  Contrast  with  the  theorem  of  §  6.  The  extraction  of  higher  roots  of 
complex  quantities  is  done  very  simply  in  terms  of  trigonometric  ratios  fsee 
Append  ixl. 


Sec.  10]  COLLEGE  ALGEBRA.  9 

The  inspection  method  of  §  6  may  be  extended  to  find  the 
square  roots  of  certain  complex  numbers  a  +  ^  1/  —  1.  Thus  for 
-3+4  V^^,  set 

r  —  3  +  2  V  —  4:=  Vx-{-  V  —  y       {x  and  y  positive). 
.  •.     -3  +  2  |/"^^  :=^x  -  y  +  2  V  —  xy. 

x  —  y—  —  Z,     xy  —  ^  (by  §9). 

Hence  :?:  =  1,  ?/  =  4,  so  that  one  square  root  of  —  3  +  4  V—  1  is 
1  +  2  s/~^\. 

EXERCISES. 
Express  with  rational  denominators 

^    2-  ya  ^     i^5  +  3  i/3  +  i/5 

3. 


2+  |/3*  *    |/5  -2*  '     1  -  |/15  * 

i/3  +  |/5  +^i/iO  ^  1  +y6  3  +  i/~^r5 

6. 


7. 


V3  -  1^5  +  VlO*  *    i/2  -  1^3  +  |/5  *  *  2  +  4/  -  1* 

3  +  5  V^^  <?  +  6  4^^^^ 

2-3  |/^^1*  '  a  -  6  4/"^* 


9.  Approximately,  ^2  =^  1.4142,  V3  =  1.7320;  find  -^ =r,     "^  "^  ^  ^  . 

4/2-1/3       1  -  4/3 

10.  Simplify  4/45  +  4/2O  +  3  4/5,      7  i/24  -  2  4/^,      4/15  --  ^^25. 

11.  Simplify  7  4/"^=^^  -  2  i\/~^^~Wl,  \/'~^y.  V~^^^^~48,  4/"^=^--  4/"3T6 

Express  in  terms  of  surds  the  square  root  of 

12.  12  -  64/3.  13.  28  -  4/300;  14.  16-8  4/3. 
15.  29  +  6  4/22.              16.  75  +  12  4/21.  17.  47-4  4/33. 
18.  «  +  6  +  V^2(Z&  +  5'^.                    19.  1  -^a"-  +  4/1  -f  a^  -f  «*. 

Express  in  the  form  <*  +  5  4/  —  1,  o^  and  6  real,  the  square  root  of 
20.  -11  +  60  4/"=n[.  21.  -  47  +  sT^^^m,, 

22.   -20+48  f/~=nr.  -  23.   -7  +  24  4/"^^. 

24.  c2  -  (^2  _  2  4/  _  c2(?^  25.  4c^  +  2(c2  -  cf 2)  ^"ZTi. 

26.  Prove,  as  in  §  2,   that  4/7  and  4/4  are  not  expressible  as  rational 
numbers. 

27.  Prove  that  1  +  4/2  is  not  a  surd  by  showing  that  an  equation 

(1  +  4/2)'*  =  r  r=  a  rational  number 
would  require  (1  —  s/%Y  =  r,  whereas  the  two  equations  are  contradictory 
since  1  +  4/2  >  1  -  4/2. 


CHAPTER  11. 
EXPONENTS;   LOGARITHMS. 

11.  If  m  is  a  positive  integer,  d^  denotes  the  product  of  m 
factors  each  of  which  is  a.  Similarly,  if  71  is  a  positive  integer, 
a"^  z=z  a  .  a  .,,  a^  to  n  factors.  Hence  d^  ,  a""  =  a  .  a ,  . ,  a  to  m -\-  n 
factors  =  «"*  +  ''.  We  may  therefore  state,  for  the  case  of  positive 
integral  indices  m,  n, 

The  First  Law  of  Indices.  The  index  of  the  product  of  tzvo 
poivers  of  the  same  quantity  is  the  sum  of  the  indices  of  the  factors  : 

(1)  a"^  .  fl^'^  =  a"*  +  ". 
As  a  corollary,  we  derive  the  formula 

a^.a^.  ^^.  .  .  .  a«=  «"*  +  "+^  +  ---  +  «. 

12.  For  the  division  of  two  positive  integral  powers  of  «,  a.^  0, 

a"^  _a  .  a,  a.  .  ,  a     (to  m  factors) 
a"       a  ,  a .  a  .  .  ,  a     (io  n  factors) 

^=  a .  a  , . ,  a     {to  m  --'  n  factors) 
if  m  >  n.     We  may  state,  for  m  and  ?^  positive  integers,  m  >  n, 

The  Second  Law  of  Indices.  The  index  of  the  quotient  of  two 
powers  of  the  same  quantity  is  the  excess  of  the  index  of  the  numer- 
ator over  the  index  of  the  defiominator  : 

(2)  «V«^  =  a"^ - "  {a  ^0,  m>n), 

13.  If  m  and  n  are  positive  integers,  we  have,  by  definition, 

[a'^Y  ^za""  .(f,,.  a"^     (to  n  factors) 

^  ^m  +  m  +  ...  +  m   -.  ^mn        (by  §11). 

Hence,  for  positive  integral  indices,  we  may  state 


Sec.  14]  COLLEGE  ALGEBRA,  ii 

The  Third  Law  of  Indices.  The  index  of  the  nth  power  of  a^  is 
the  product  of  the  indices  m  and  n:  -        - 

(3)  (pry  =  a^. 

14,  We  next  extend  the  use  of  the  symbol  a''  to  cases  in  which 
n  is  negative  or  fractional^  assuming  that  such  new  symbols  d^  will 
satisfy  certain  cases  of  the  above  first  law  of  indices,  and  proceed 
to  determine  what  meaning,  if  any,  may  be  attached  to  the  gen- 
eralized symbols.  It  is  later  shown  (§  15)  that  the  symbols,  with 
the  meanings  thus  obtained,  satisfy  the  three  laws  of  indices  for  m 
and  n  any  rational  numbers.  For  this  reason  the  interpretations 
of  the  symbols  are  justified  and  the  desired  permanence  of  the 
algebraic  laws  is  attained. 

Consider  the  symbol  a-^,  where  q  is  any  positive  integer.  Since 
the  symbol  shall  satisfy  the  first  law  of  indices, 

a^  .a^.a^...{toq  factors)  =  a^  +  ^^-^  +  "-^^o,terms)  ^  ^i  ^  ^^ 

Hence  a  «  must  be  such  that  its  ^th  power  is  a,  that  is,* 

a'i  =  ya. 

Similarly,  the  symbol  a^^*^,  where  ;;  and  q  are  positive  integers, 

must  be  such  that 

p       p  p     p 

aJ .  aJ. .  .  (to  g  f actors).  =  ^i^-^  +  y  ^•••(*°«^®^^'^>  _  ^p^ 

so  that  a^^^  must  be  a  ^th  root  of  a^.     Hence 


1 

Since  the  symbol  a'l  obeys  the  first  law  of  indices,  we  find  that 

a «  .  a 5 .  .  .  (to  jt?  factors)  =  ci'^  z=z  (^  Va  )  . 

*  The  radical  sign  is  used  to  denote  a  particular  root.     Thus 
|/4  =  +  2,  -  4/4  =  -  2;  hence  4^  =  +  2. 


12  EXPONENTS;    LOGARITHMS,  [Ch.  II 

Hence  the  two  resulting  values  for  a^^''  must  be  equal, 

a  true  theorem  on  radicals.     Thus  VS'^  =:  \  yS  )   =4. 
Consider  the  symbol  a^.     By  the  first  law  of  indices, 

so  that  a^  =  a!'/oJ'  ==1. 

Hence,  if  a  be  any  number  different  from  zero,  a^  —  \, 

Consider  the  symbol  a  ~%  where  r  is  a  positive  integer  or  frac- 
tion, and  assume  that  it  may  be  combined  with  the  symbols  already 
defined  in  such  a  way  that  the  first  law  of  indices  holds.     Then 

a-^'-dr  z=.ar-''  ^  a'=  1, 

or 

Hence  a'""  denotes  the  reciprocal  of  oT, 

As  examples,  9^^  r=  VO^  =  I/36  =.33  =  27, 

4  "  *  =  1  -^  43/2  ^  1  ^  ^743-^  1/g^ 

(  _  27)"  ^  =  1  ^  (  -  27)*  =  1  -^  (  ^  -  27)'  =  1  -^  (-  3)2=  1/9. 
15.  We  have  been  led  to  the  following  definitions: 

(Def.)  a^=  V~a^,    aP  =  1,    a-^^^  =  1  -^  a^^S 

so  that  we  hav6  a  definition  for  a""  for  every  rational  number  n. 
In  order  that  these  generalized  symbols  a^,  a^  shall  prove  to  have 
practical  value,  it  is  essential  that  they  shall  obey  the  three  formal 
laws  of  indices  which  have  been  proved  to  hold  for  positive  integral 
exponents  (§§  11,  12,  13).  In  proving  that  this  is  the  case,  we 
make  use  of  certain  properties  of  radicals,  namely: 

(A)  vT=  7T'. 

(B)  i/J-  Vt  =  Vt^i, 

(c)  \^i^  ^r=  VUi' 


Sec.  15] 


COLLEGE  ALGEBRA, 


13 


I.  To  prove  that  dJ^dP-  =  0^  +  ""  whe7i  m  and  n  are  rational 
numbers. 

(1)  Let  m  and  n  each  be  a  positive  fraction,  including  the  case 
in  which  the  denominator  is  unity.     Then 


«««»  =  Va''j/a'- 

(Def.) 

=  1/a'«  y^"^ 

(by  A) 

=  Ya"'  ■  a"! 

(byB) 

~   -"^a^'  +  rq 

(by  first  law,  §  11) 

ps+  rq 

=z  a    ^^ 

(Def.) 

P.+  Z 
=  a^     «/ 

(2)  Let  m  be  a  positive  fraction  —  and  n  a  negative  fraction 


^     T  p      r 

I*—  — ,  and  let  —  >  — .     Then  ps  —  rq  is  positive. 


a'ia    «  —  \/a^  -^  \/a'' 


=  ?^ 


ps  — rq 


,,so  that  a'^a''  =  «'^  +  ''  in  this  case. 

(3)  Let  m  =  ^,  7^  =  -  -,  with  ^  <  -. 
q  s  q       s 


(Def.) 
(by  A) 

(byC) 

(by  second  law,  §  12) 
(Def.) 


By  case  (2) 


a^a~^  =  a^~  ^ 


(since  -  >  -j. 
\  s       qj 

Taking  the  reciprocal  of  each  member,  we  have  by  (Def.), 


14.  EXPONENTS;    LOGARITHMS.  [Ch.  II 

P  T  P     T 

(4)  Letm  =  —  — ,  7^  = ,  where  — ,  -  are  positive  fractions. 

^  ^  q  s  q    s  ^ 

Taking  the  reciprocal  of  the  two  equal  expressions  in  case  (1)  and 

applying  the  definition  of  a  negative  exponent,  we  get 

a    9a    ^  =  a    ^«     sy  z=  a    «     «. 

(5)  Let  finally  m  be  rational  and  7i  be  zero.        By  (Def.) 

II.  To  prove  that  a^  -^  d^  =  aJ^"^  ivlien  m  mid  n  are  rational, 

a'»-^a™  =  fl^^".  —  -  a"*.  a-«      (Def.) 

since  m  and  (—  n)  are  rational  numbers,  so  that  Theorem  I  applies. 

III.  To  'prove  that  {a'^'Y  =  ^"'''  when  m  and  n  are  rational, 

(1)  Let  m  be  a  positive  fraction  and  n  a  positive  integer.    Then 


Va«/  =  f  a^.f  aP.  .  .  (to  n  factors) 

(Def.) 

=  \/a^'a^ ...  (to  n  factors) 
=  \/a^ 

(by  B) 

(§  11,  corollary) 

(2)  Let  w  and  n  be  positive  fractions.     Then 

(Def.) 
(Def.) 

=  \'ai 

[by  case  (1)] 

pr 


=  ««*  (Def.) 

(3)  Let  w  be  a  positive  and  n  a  negative  number,  integral  or 
fractional.    Then,  using  the  definition  of  a  negative  exponent  twice, 

^""^^'^  "  7^-"  ^^  ^  ^~""         fby  case  (2)1. 

(4)  Let  m  be  negative  and  n  positive.     Then 


(«-")" =(i^r=(;^»' 


Sec.  15]  COLLEGE  ALGEBRA.  15 


also  for  n  a  positive  fraction  in  view  of 


since  [—     =  ~n'^^  ^^"^^  ^^^"  positive  integral  n  by  inspection,  and 


\WI  ^      \l0l  to''  ^^f^r  ?^^/^ 

Next,  {a^y^  =  a^'\  for  /i  and  n  positive,  by  case  (2).     Hence 

(5)  Let,  finally,  m  and  n  be  both  negative.     Then 

ia-^V*"  —  z r  —  — —  [by  case  (4)1 


=  a^\  (Def.). 

e  laws  of  indices  therefore  hold  when  the  exponents  m  and  n  are 
any  rational  numbers.  As  an  exercise,  the  student  may  prove  that, 
for  any  rational  number  7^, 

By  way  of  illustration  of  the  laws  of  indices,  we  note  that 

3  1  3  1  5  3  1  13 


^H       Express  with  positive  indices 

7.  3a- I  8.    2-^0^-2.  9.  S^Ba?"*. 

10.  2~^G~^  11.    V  x^  -^Vx~^' 


1  2        i  1  8    .1      1 

a"  a      'a'  =  a'      ~''^  '  z=z  a^  =  1. 

EXERCISES. 
Find  the  numerical  value  of 

S\        2.  16     '.         3.  9    \         4.    \g)       .         6.  25     '.        6.  lOO"" '. 


l6  EXPONENTS;    LOGARITHMS,  [Ch.  I] 

Express  by  radical  signs  with  positive  indices 
12.  al  13.  J.  14.  a.  16.  a~^ 


16..-5«l  n.5^\  18.  /.-V.-^     19.§^\ 


2       • 

a  x' 


20.  ST.  21 


Simplify 


22 


.  a^a~%a)~K(^''y\  23.  aH'-a-^^l 

24.  (a^  l^h^  "  V  4/5  -  4  |/^F^.  25. 


a     ^6 


a*5'        '    5-^* 
26.  2^(2^-T-2"^^2"~^  27.  (  2")""^  h- (s^-O^""'. 

28.  Multiply  a;~^ +y~Hyaj~  ^-  y'^l 

29.  Multiply  a*"  +  a"^  +  1  by  a"  "^  +  a"  2  ^  1. 

30.  Divide  32aj "  ^  +  12aj"  *  +  lOaj"  ^  -  12  by  2iB-  ^  ~  1. 

31.  Factor  a^  -  &^     :?  -  y,     x~  ^  -  27y~  \ 

32.  Find  the  square  root  of  a  '*  —  Qa^  -j-  6a^-\-  12a"  +  4. 

33.  Find  the  square  root  of  4a  —  4a®  -f-  a  +  2a  -\-a^  —  4a'. 

34.  Solve  aj  -  a;^  -  30  =  0;    aj^  -  3aj*  -f  2  =  0. 

35.  Solveaj^  =  8«~^  +  2;      a;"*+ 16  =  8aJ~^ 

16.  What  meaning,  if  any,  may  be  attached  to  the  new  symbol 
3^^^ ,  in  which  the  index  4/2  is  not  rational  ?  Since  4/2  can  not  be 
expressed  as  a  fraction  (Chapter  I),  the  symbol  3*"^  is  of  a  charactei 
not  previously  considered.  But  4/2  may  be  approximated  by  a  deci- 
mal fraction  to  any  desired  degree  of  approximation.  For  example, 
4/2  lies  between  1.414  and  1.415;  it  lies  between  1.41421  and 
1.41422;  etc.     We  have  obtained  a  definite  meaning  for  the  expres- 

i4J_4.  1415 

sions  31^00  and  3  ioott.  It  is  natural  to  say  that  the  value  of 
3*'^  lie$  between  the  values  of  these  two  expressions.  As  a  closer 
approximation,  the  value  attached  to  3^^  lies  between 

141421       ,     141422 

3Ti7iyirTrir    and    31^^^^^. 


Sec.  17]  COLLEGE  ALGEBRA.  17 

Continuing  the  process  of  approximation,  we  obtain  two  fractions 

—  and  — ^^,  which  differ  by  an  amount  —  which  can  be  made  as 
n  n  n 

small  as  we  please,  and  such  that  4/2  lies  between  the  two  fractions. 
By  this  limiting  process,  tliere  is  pointed  out,  with  as  great  a 
degree  of  precision  as  we  may  desire,  a  so-called  limit,  which  we 
define  to  be  the  value  of  3*"^  .       Similar  definitions  apply  to  the 

symbols  2*'^  ,  a^^  and,  generally,  to  d^  where  n  is  any  real  number. 
A  series  for  a^  in  ascending  powers  of  n  will  be  given  in  Chapter 
XVI;  so  that  a  more  practical  method  is  derived  for  the  calculation 
of  a^ 

If  decimals  correct  to  five  places  be  used  for  3  ^^  and  3  ^ ,  the 
product  of  the  decimals  will  equal  3  ^^  +  '^^^  to  an  approximation 
correct  to  five  places.  *  It  follows  that  in  numerical  work  we  may 
employ  the  laws  of  indices  when  dealing  with  symbols  like  3*"^  by 
means  of  their  approximate  values.  We  make  the  assumption  that 
the  laws  hold  for  the  symbols  3^^  ,  .  .  .  ,  themselves. 

LOGARITHMS. 

17.  By  the  first  two  laws  of  indices,  two  powers  aP"  and  aP^  of  the 
isame  quantity  a  may  be  multiplied  or  divided  by  merely  adding  or 
subtracting  the  indices  m,  n.  Now  addition  and  subtraction  are 
[simpler  operations  to  perform  numerically  than  multiplication  and 
jdi vision.  It  would  thus  appear  desirable  to  be  able  to  express  all  real 
numbers  as  powers  of  a  single  number  a,  called  the  base.  Only 
ipositive  numbers  will  be  employed  as  bases. 

For  example,  taking  a  =  2,  we  have 

i=:2'^    8=:2.^    1  =  2^   i  =  2-^    i=2-3,    |/^=2^    4/8  =  2"'. 

*In  employing  only  five  decimals,  the  fifth  place  in  the  product  may  vary 
3y  unity  from  the  correct  result  for  the  fifth  place  in  the  most  unfavorable 
lases.  Thus,  if  only  five  places  be  retained  in  .972385,  the  square  would  be 
ncorrect  by  more  than  .000009,  which  would  be  replaced  by  .00001  when  only 
ive  places  are  retained. 


l8  EXPONENTS;    LOGARITHMS.  [Ch.  Il'i 

But  we  do  not  find  by  inspection  the  power  of  2  which  will  give 
certain  numbers,  for  example,  3.     By  trial,  we  find  that 

9  3  8  3  9  10 

2  <  26"  <  22 4  <  3  <  22  4  <  2~6-  <  22. 

38 

Now  22  4  —  2.996  to  three  decimal  places.  The  required  exponent 
is  slightly  greater  than  f  f ,  but  is  less  than  ||.  Just  as  the  symbol 
/j^3  was  introduced  to  denote  the  real  positive  root  of  the  equation 
x^  =  3,  so  now  we  employ  the  symbol  'logg  3,  read  logarithm  3  base 
2,  to  denote  the  exponent  x  which  makes  2"^  =  3. 

Definition. — The  exponent  of  the  poiver  of  the  base  a  luhich  gives 
rise  to  a  oiumher  N  is  called  the  logarithm  of  N  to  the  base  a; 
symholically , 

(4)  iV^i=:a^«^a^. 

Thus,  for  a  =  2^  the  above  relations  give 
2  =  log,  4,    3=  log,  8,    O  =  log,l,    -l=--log,i,    -3  =  log,i    j 
i  =  logy2,     f-log,4/8,     IK  log,  3  <  If.  ' 

Similarly,  1000  =  10^  gives  3  =  log,,  1000; 
.01  =  10- 2  gives  -  2  =  log,,  .01. 
18.  For  JV  =■  a,  equation  (4)  gives  log^«  =  1.      Also  a^  —  1 
gives  loga  1  =  0.     Hence,  whatever  the  base  may  de,  the  logarithm 
of  1  is  zero  and  the  logarithm  of  the  hase  is  unity. 

Since  logarithms  are  exponents,  the  three  laws  of  indices  (ex- 
ponents) give  rise  to  three  corresponding  properties  of  logarithms. 

I.  The  logarithm  of  a  product  equals  the  sum  of  the  logarithms 
of  its  factors. 

Let  the  factors  be  N  and  M.     By  the  definition  (4)    we  have 

(5)  iV=a^«^a^,     M  =  a'^'^a^. 

Hence,  by  the  first  law  of  indices,  and  the  definition  (4), 

.  •.     logaNM  =  log^JV  +  logJL 

II.  The  logarithm  of  a  quotient  equals  the  logarithm  of  the 
divide7id  minus  the  logarithm  of  the  divisor. 


Sec.  19] 


COLLEGE  ALGEBRA. 


19 


Let  the  dividend  be  N  and  the  divisor  M,    Applying  the  second 
law  of  indices  to  the  identities  (5),  we  get 

M 

N 
By  the  definition  (4)  applied  to  the  number  -^,  we  get 

N 


M 


N 


\0gaM. 


III.  The  logarithm  of  the  ptli  -poiuer  of  a  number  equals  p  times 
'he  logarithm  of  the  numler. 

Let  iV^  be  the  number.  Raising  the  two  members  of  (4)  to  the 
30wer  p  and  applying  the  third  law  of  indices,  we  get 

3ut,  by  definition  (4)  applied  to  the  numiber  iV^, 

.-.    ^log,iV^^=7^1og,iV. 
19.  Compare  the  logarithms  to  the  base  2  which  are  given  in 
17  with  the  logarithms  of  the  same  numbers  to  the  base  8: 
=  log84,    l^loggS,  O^loggl,  -i^logsi,    -1  =:=log8|-,  ..  . 
¥e  observe,  for  each  number  iV,   that  logg  N  =  \  logg  N,      But 
=  logg  2.     Hence  there  is  a  constant  multiplier  logg  2  in  passing 
rom  logarithms  to  the  base  2  to  logarithms  to  the  base  8. 
The  student  may  readily  verify  the  results  in  the  table 


Number  N  . 

Log^io^ 

Log.oo^ 

Log^.^iV 

10,000 

4 

2 

12 

1,000 

•3 

3/2 

9 

100 

2 

1 

6 

10 

1 

1/2 

3 

1 

0 

0 

0 

0.1 

- 1 

-1/2 

-  3 

0.01 

-  2 

-     1 

-  6 

0.001 

-3 

-  3/2 

-  9 

20  EXPONENTS;    LOGARITHMS,  [Ch.  II 

For  the  numbers  iV^of  the  table,  logiooJ^^=  i  l^gio^j  t^G  multiplier 

which  converts  logarithms  to  the  base  10  into  logarithms  to  the 

base  100  is  ^  =  logjoolO.      Similarly,   a  logarithm  to  the  base  10 

must  be  multiplied  by  3  =  logio^  10  to  give  the  logarithm  of  the 

t 
same  number  to  the  base  10^. 

In  general,  suppose  that  the  first  base  is  a  and  that  the  second 

base  is  b.     By  the  definition  (4), 

.  •.        h  l«g/  z=z  a  ^^^a^  =  {b  ^°^6")  ^^^a^. 

Applying  the  third  law  of  indices,  we  get 

logt,]Sr=]ogaN.  logf,a. 

To  pass  fro7n   a  table  of  logarithms  to  the  base  a>  to  a  table  of 

logarithms  to  the  base  b,  we  employ  the  constant  multiplier  log^  a,  th6\ 

logarithm  of  the  old  base  with  resyect  to  the  neio  base, 

EXERCISES. 

1.  Find  log2  32,     log^  32,     logi  2  |/8,     logi  ^64,     logs  5  i/5. 

2.  logc  h  .  logb  N  =  logc  a  .  loga  J^. 

3.  loga  J^  -^  logo  ^  =  l0g6  J^  -^  logb  M. 

4.  Express  logio  |,  logio  60,  logio  450  in  terms  of  logio  2  and  logio  3. 
6.  Solve  52-»  -  5^  + 1  +  6  =  0  in  terms  of  logio  3  and  logio  3. 

6.  Simplify  log  (  |/144  |/i08  -f-  1/1728"  \/m). 

20.  Logarithms  to  the  base  10  are  known  as  Common  Loga- 
rithms* and  are  used  in  numerical  calculations.  The  chief  advan- 
tage of  the  base  10  lies  in  the  fact  that  from  a  known  logarithm 
of  a  number  ^may  be  derived  by  inspection  the  logarithms  of  all 
numbers  differing  from  iV^  only  in  the  position  of  the  decimal  point. 
Thus,  if  we  know  that  logio  2-23  =  .3483,  then  logio  22.3  =  log^olO 
4-  logio  2.23  ==  1.3483,  and  logio  2230  ^  3.3483.  The  decimal 
part  .3483  common  to  the  three  logarithms  is  called  tlie  mantissao 
The  integral  part  of  a  logarithm  is  called  the  characteristic;   it 

*  Introduced  in  1615  by  Briggs  (1556-1630),  a  contemporary  of  Napier 
(1550-1617),  the  inventor  of  logarithms.  For  a  history  of  logarithms  see  the 
article  by  Glaisher  on  Logarithms  in  the  Encyclopaedia  Britannica. 


Sec.  21]  COLLEGE  ALGEBRA.  21 

depends  upon  the  position  of  the  decimal  point  in  the  given  number. 
Thus  logio  2.23,  logio  22.3,  logio  2230  have  respectively  the  charac- 
teristics 0,  1,  3. 

21.  That  a  shifting  of  the  decimal  point  in  any  number  iV^does 
not  alter  the  mantissa  of  logio  ^  follows  from  the  fact  that  the 
shifting  of  the  decimal  point  p  places  to  the  right  multiplies  iV^by 
10^,  the  shifting  q  places  to  the  left  divides  iYby  10^.     But 

loglo(i^^x  10^)  ==  \og,,N  +  p\og^^  10  =  i?  + logio  iV, 
j  logio  (^-^  10^)  =  logio  N-q  logio  10  =  -  ^  +  logio  ^ 

In  determining  the  mantissa  we  may  tlierefore  disregard  the  decimal 
point  in  the  number.  Thus,  for  the  number  22.3,  we  use  the 
sequence  of  digits  223  in  looking  for  the  mantissa  in  the  Table  of 
Logarithms  given  below.  In  taking  the  mantissa  from  the  Table, 
the  decimal  point  is  to  be  placed  before  the  digits  found.  Hence 
for  logio  22.3  the  mantissa  is  .3483;  by  inspection,  22.3  lies  between 
10^  and  10^  so  that  the  characteristic  is  1.    Hence  logio  22.3  =  1.3483. 

22.  We  may  determine  the  characteristic  of  the  logarithm  of 
my  number  iV^by  inspection.  The  place  immediately  to  the  left  of 
the  decimal  point  is  known  as  the  units  place.     If  there  be  digits  to 

,Lhe  left  of  units  place,  let  their  number  be  p.  Then  N  =^  M  X  10^, 
vvhere  if  is  a  number  whose  first  digit  is  in  units  place,  so  that  M 
.ies  between  10  and  1  =  10^  The  characteristic  of  logio  if  is  evi- 
lently  zero  and  therefore  that  of  logio  ^  is  p.  But  if  the  first  signif- 
cant  digit  in  iV^is  in  the  nih  place  to  the  right  of  units  place,  then 
N=  M  --  10%  if  being  defined  as  before.  Thus,  if  JSf  =  0.0024, 
^  =  3  and  M  —  2.4.  Then  the  characteristic  of  logio  iV^  is  —  ^^. 
iVe  may  combine  the  results  to  give  the  theorem : 

The  char  act  eristic  of  the  logarithm  of  a  number  is  -\-  n  if  the 
first  significant  digit  lies  n  places  to  the  left  of  units  place,  but  is 
—  nif  the  first  significant  digit  lies  n  places  to  the  right  of  units 
ilace. 

Example.  Find  the  logarithms  base  10  of  2400,  .24,  and  .0024. 

From  the  Table,  we  get  log  2.4  =  .3802.     Hence 

log  2400  =  3.3802,     log  0.24  =T.3802,     log  0.0024  =  3.3802. 


22  EXPONENTS;    LOGARITHMS,  [Cn.  II 

As  in  this  example,  the  negative  sign  belonging  to  the  charac- 
teristic is  written  above  the  latter.  Otherwise,  —  1.3802  would  bj 
taken  to  mean  that  both  1  and  the  decimal  .3802  were  negative; 
whereas  the  mantissa  is  understood  to  be  always  positive.  Hence 
1.3802  denotes  -  1  +  0.3802.  If  we  desired  to  find  log  ^/.'M,  m 
should  divide  log  .24  ==  1.3802  by  2.     To  do  this,  we  write  i 

i(T.3802)  =r:  |(  -  2  +  1.3802)  =  -  1  +  .6901  =  T.6901. 

23.  There  follows  on  pages  24,  25  a  table  of  Logarithms  to  the  bas< 
10  correct  to  four  decimal  places.     For  its  origin  see  Chapter  XVI 

24.  To  find  hy  the  TaUe  the  logaritlim  of  a  given  numter.' 
Suppose  the  given  number  to  consist  of  three  digits,  as  2.23.    Th« 

first  two  digits  22  are  to  be  found  in  the  left-hand  column  headed  i\^ 
On  a  line  with  these  and  in  the  column  headed  by  the  third  digit  3i 
we  find  3483.  With  the  decimal  point  prefixed,  the  result  .3483  i 
the  mantissa  of  log  2.23.  Since  there  are  no  places  to  the  left  o; 
units  place,  we  get 

log  2.23  =  0.3483. 

Similarly,  log  7  =  log  7.0  ==  0.8451. 

If  the  given  number  consists  of  four  or  more  digit^Has  2.2325 
we  determine  the  logarithms  of  the  numbers  of  three  digits  whicl 
are  respectively  less  than  and  greater  than  the  given  number.  Thu 
log  2.2300  =  0.3483 
log  2.2400  z=  0.3502 
The  logarithm  of  2.2325  lies  between  these  two  logarithms,  whost 
difference  is  .0019.  But  2.2325  exceeds  the  smaller  number  2.230( 
by  y%'V  of  the  difference  of  the  two  numbers.  Hence  log  2.232i 
must  exceed  log  2.2300  by  approximately  -f-^-^  of  .0019,  so  tha 
log  2.2325  =  0.3483  +  .0005  =  0.3488.t 

*  Henceforth  the  base  is  supposed  to  be  10. 

f  The  value  .0005  is  used  for  .000475,  since  the  former  is  the  decimal  of  fou 
places  nearest  to  the  latter.  It  would  be  deceptive  to  retain  more  than  fou 
places,  since  the  results  from  the  Table  are  only  true  to  four  places,  and  th 
retention  of  six  places  would  indicate  our  belief  in  the  accuracy  of  the  fifti 
and  sixth  places. 


Sec.  25]  COLLEGE  ALGEBRA,  23 

In  Chapter  XVI,  we  establisli  the  principle  here  involved, 
namely,  /or  a  relatively  small  increase  in  a  numler^  its  logaritlijn 
increases  proportionally.     For  example,  from  the  Table : 

log  9.34  =  .9703,  log  9.35  =  .9708,  log  9.36  =  .9713, 
log  9.37  =  .9717,  log  9.38  =  .9722,  log  9.39  =  .9727, 
the  difference  between  adjacent  logarithms  being  5  in  four  cases 
and  4  in  one  case.  The  difference  1  in  the  numbers  934,  935 ,  .  .  .  , 
939  is  relatively  small.  On  the  contrary,  the  principle  is  not 
valid  if  applied  to  the  numbers  100,  200,  300,  whose  differences  are 
not  relatively  small.     Thus,  from  the  Table, 

log  100  =  2.0000,     log  200  =  2.3010,     log  300  =  2.4771. 

25.  To  find  hy  the  Table  the  number  corresponding  to  a  given 
logarithm. 

Let  the  given  logarithm  be  3.9779  and  the  corresponding  number 
be  xY.  Since  the  mantissa  .9779  does  not  occur  in  the  Table,  we 
take  those  numbers  in  the  body  of  the  Table,  viz.,  9777  and  9782, 
which  are  just  larger  and  just  smaller  than  9779,  respectively.    Then 

log  9500  =  3.9777  1  j.  .0002  =  difference 
log  N  =  3.9779  y  \0005  =  difference 
log  9510=:  3.9782 

Hence  the  difference  between  9500  and  iV^must  equal  f  .of  10,  the 
difference  between  9500  and  9510.  Hence  J^  =  9500  +  4  =  9504. 
If  the  given  logarithm  is  3.1795,  the  work  is  as  follows: 

log  .00151  =  3.1790 
log  JV  '  =  3.1795 
log  .00152  =  3.1818 


|.  .0005  =  difference 
*  .0028  =  difference 


Hence  the  difference  betw^een  .00151  and  iVmust  equal  2\ot  .00001, 
the  difference  between  .00151  and  .00152.  Hence  J^  =  .00151 
+  .000002  =  .001512  to  four  significant  places.  /For  2^  x  .00001 
=  .00000178  +,  we  take  .000002  as  the  additive  part  to  obtain  iV, 
being  a  decimal  all  of  whose  figures  are  reliable.    Had  we  used  a  six- 


24 


EXPONENTS;    LOGARITHMS. 
TABLE   OF  FOUR-PLACE   LOGARITHMS. 


[Ch.  II 


N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

0000 

0000 

3010 

4771 

6021 

6990 

7782 

8451 

9031 

9542 

1 

0000 

0414 

0792 

1139 

1461 

1761 

,2041 

2304 

2553 

2788 

2 

3010 

3222 

3424 

3617 

3802 

3979 

4150 

4314 

4472 

4624 

3 

4771 

4914 

5051 

5185 

5315 

5441 

5563 

5682 

5798 

5911 

4 

6021 

6128 

6232 

6335 

6435 

6532 

6628 

6721 

6812 

6902 

5 

6990 

7076 

7160 

7243 

7324 

7404 

7482 

7559 

7634 

7709 

6 

7782 

7853 

7924 

7993 

8062 

8129 

8195 

8261 

8325 

8388 

7 

8451 

8513 

8573 

8633 

8692 

8751 

8808 

8865 

8921 

8976 

8 

9031 

9085 

9138 

9191 

9243 

9294 

9345 

9395 

9445 

9494 

9 

9542 

9590 

9638 

9685 

9731 

9777 

9823 

9868 

9912 

9956 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2830 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

■  3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

•4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

88 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

COLLEGE  ALGEBRA. 
TABLE   OF  FOUR-PLACE   LOGARITHMS. 


25 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

50 

6990 

6998 

7097 

7016 

7024 

7033 

7042 

7050 

7059 

.7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177  • 

7185 

7193 

7202 

7^10 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

"8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

950.4 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9052 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9V82 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

26  EXPONENTS;    LOGARITHMS.  [Ch!  H 

place  table  of  logarithms,  the  additive  part  would  have  been  retained 
in  the  form  .00000178. 

26.  An  equation  in  which  the  unknown  quantity  appears  in  an 
exponent  is  called  an  exponential  equation.  Such  equations  may 
usually  be  solved  with  the  aid  of  logarithms. 

Example.  Find  the  value  of  a?  in  3^^  +  ^  =  3^^~*. 
Taking  the  logarithm  of  each  member  of  the  equation,  we  get 
(2  a?  +  1)  log  3  =  (bx  -  4)  lo^'g  2. 
.-.     a;  (2  log  3  -  5  log  2)  =  -  4  log  2  -  log  3. 
.  •.     X  (log  3^  -  log  2"^)  =  -  log  (2*  X  3). 
,      ^  ^        -  log  4_^  ^    -  16812  ^  L6812  ^  ^^^^^^ 
log  9  -  log  32  1.4491  .5509 

to  four  decimal  places.     The  denominator  may  be  calculated  otherwise,  since 
it  equals 

log  -5  =  log  ?l^  =  log  (.28125)  =  1.4491. 
2  10  " 

EXEKCISES. 

1.  Find  the  common  logarithms  of  6752,  5.4321,  0.04682. 

2.  What  numbers  have  the  logarithms  2.1516,  T.2222,  43333  ? 

3.  Solve  3^-'  -  450,  lO'"'^  :=  4'"'^,  3^  +  1  s^^-^  ^  1000. 

Using  logarithms,  calculate  to  four  significant  places : 
8124  X  .00345 


*•  .-00069^8-7X2-  5.t/l2345.-      6.(84.625).. 


100,—        10,- 


7.    V45.24  X  1^.0004  X  |/12345.       8.  2T  9.    1/5  X  |/8765. 

Solve  by  logarithms  the  simultaneous  equations: 

10.  3'^  +  '^==  10,  5^-'  =  125^'^-'.  11.  3^  +  ^  =  6'',  2^  =  9  X  2^-\ 

12.  Find  the  number  of  digits  in  25^^,  in  2*T 


CHAPTEE   III. 

FACTOR  THEOREM;     QUADRATIC   EQUATIONS. 

1       27.  It  is  observed  that  x  —  a  is  a  factor  of 

x^  —  d\     x^  ■—  ^ax-{-  2a^,     x^  —  ax^  —  x  -f  «, 
and  tliat  eacli  expression  vanishes  when  a  is  written  in  place  of  x. 
In  general,  if  an  expression  E^  involving  x  has  a  factor  x  —  a,  and 
if  Q^  is  the  quotient  arising  from  the  division  of  B^  by  x  —  a,  then 

E^  =  {x-  a)Q^, 
so  that  E^  vanishes  when  a  is  written  in  place  of  x.     The  inverse 
of  this  theoi^em  is  called 

The  Factor  Theorem.     If  an  integral  expressio^i  involving  x 
vanishes  tulien  a  is  iuritte7i  in  place  of  x^  it  lias  ill e  factor  x  —  a. 

Divide  the  given  expression  E^hj  x  —  a  until  a  remainder  R  is 
obtained  which  does  not  involve  x.     If  Q^  be  the  quotient,  then 

E^,=  {x-a)Q,+  R. 
Substituting  a  for  x  in  this  identity,  we  get 

where  E^  is  the  value  of  E^  when  a  is  w^ritten  for  x.  By  hypotliesis, 
E^  vanishes,  so  that  the  remainder  R  is  zero.  Hence  E^  is  exactly 
divisible  by  x  —  a. 

Incidentally,  we  have  also  established 

The  Remainder  Theorem.     Upon  dividiyig  an  integral  expression 
E-^  iy  X  —  a,  we  oUain  as  the  remainder  Ea-,  the  value  of  E^  when 
\j^ki^  written  for  x. 

^f  Example  1.  To  verify  tliat  x  -  a  is  a  factor  of  a:r>  —  a^,  we  put  a  for  x  in 
tlie  expression  and  get  ap  —  aP  —  0. 

Example  2.  a?  —  1  divides  :i?  --2x^  -\-\,  since  1  —  2-1-1  =  0. 

27 


28  FACTOR   THEOREM;    QUADRATIC  EQUATIONS,      [Ch   III 

28.  We  next  apply  the  factor  theorem  to  prove  tliat,  if  n  be  a 
positive  whole  number,  x"^  —  y'^  is  always  divisible  by  x  —  y^  but 
that  x""  +  y""  is  never  divisible  hj  x  —  y.  Putting  y  in  place  of  x, 
the  expressions  become,  respectively, 

To  prove  that  x""  +  V""  has  the  factor  x  +  y  when  n  is  odd,  but 
not  when  n  is  even,  we  note  that  x-\-y  —  x—  {  —  y).  Upon 
writing  —  ^  in  place  of  x^  x"^  +  2/''  becomes  (  —  l)*y  +  y"^,  which 
vanishes  if  n  is  odd,  but  not  if  7i  is  even.  Similarly,  x"'  —  y"^  has 
the  factor  x -\-  y  ii  n  \^  even,  but  not  if  n  is  odd. 

In  particular,  r''  —  1  is  always  divisible  by  r  —  1.     We  have 

r^  -\  =  (r  -  l){r  +  1), 

^3- 1  =  (r- l)('r2  +  r+ 1), 

r^  _  1  =  (r  -  l)(r3  +  r^  +  r  +  1). 

To  establish  the  general  formula 

(1)  r''-l  =  {r  -  l)(r^-'  +  r"-2  +  .  ..  +r  +  l), 
we  perform  the  multiplication  on  the  right  and  obtain 

(^n  _|_  ^n-1  _|_  _  ,    _|_  ^2  _^  ^)    _    (^.n-1  _|_  _  ^  _|_  ;.  _|_  1)    :=^  ;.n  _   1^ 

Substituting  -  for  r  in  (1)  and  multiplying  by  y'\  we  get 

(2)  x""-  y'^={x-y){x''-^ ^x""-^ y-^x^-^ y'^^ .  .  .+ ^^"~'+^"'*')» 
Eeplacing  y  by  —  y  we  obtain,  //*  -^^  15  odd, 

Eeplacing  ?/  by  —  i/  in  (2),  we  obtain,  if  n  is  even, 
(4)  x""-  y''^  {x  +  y){x''-'-  x^'-hj+x'^-^y'^ - .  .  .+  xy^'-^-y''-'). 

89.  The  preceding  formulae  may  be  applied  to  find  a  factor 
which  will  rationalize  any  binomial  surd.     For  example, 

^3-  1  ~  (^3"-  OCt'B  +  f/3  +  1)        '       3-1 


Sec.  30]  COLLEGE  ALGEBRA,  29 

If  the  given  surd  \^  x  —  y,  where  x  =  \/a  and  y  =  ^h,  then  a;" 
and  y"^  are  rational  if  n  be  a  multiple  of  both  p  and  q.  Applying 
formula  (2),  we  see  that  the  rationalizing  factor  is 

the  rational  product  being  x'''  —  ^^ 

If  the  given  surd  is  x  +  y,  where  x,  y,  n  are  defined  as  above, 
then,  for  n  odd,  formula  (3)  gives  the  rationalizing  factor 

and  the  rational  product  x^ -\- y'^]  wliile  for  n  even,  the  rational 
product  is  x"^  —  ?/"  and,  by  formula  (4),  the  rationalizing  factor  is 

x^-y-^x^-hj  -\-x^-hf  —  ...  -\-xif-'^  —  y""'^* 
For  example,  to  rationalize  |/7  +  y  5,  the  rationalizing  factor  is 

(/7  )'-G/ry  i/b  +  (4/7  )'  ^25"-  (/f)'  .  5+4/7".  54/^-  5^25" 
and  the  rational  product  is  (4/?  )^  —  (t^S  )^  =  7^  -  5^  =  318. 

RXEHCISES. 

Decompose  into  three  or  more  real  factors 

1.  ^*  -  y\  2.  x^  -  y.  3.  x  +  /.  4.  oj^  -  /. 

Without  actual  division  show  that 

5.  18a?^^  +  19^'^  f  1  is  divisible  by  ic  +  1. 

6.  2«*  -  i^^  -  6aj^  +  4a;  -  8    is  divisible  by  ^  -L      ' 

7.  x^  -  ?>x  -f-  3ic^  -  3a?  +  2  is  divisible  by  a;^  -  3ic  +  2. 
Without  actual  division  find  the  remainder  when 

8.  x^  -^x  ^-x-^  is  divided  by  a;  +  3. 

9.  x'  -f  %xy^  -f-  y^      is  divided  by  cc  —  2y. 
Express  with  rational  denominator 

1  v/  1  1  +  |/r  4^2"  i/9" 

4/3  -  V5  V'2  -1-1  1  -  1/4  4/2  +  4/3 

14.  {x  -\-  yf  —  a;'''  —  y^  and  (a;  -f  y^  —  a?^  —  ?/^  are  divisible  by  a;  -\-  xy-\-y^ 

30.  A  quadratic  equation  is  an  equation  wliich  involves  the 
square,  but  no  higher  power,  of  the  unknown  quantity.     When  all 


I 


3©  FACTOR    THEOREM;    QUADRATIC  EQUATIONS.      [Cu.  Ill 

the  terms  have  been  transposed  to  the  left  member  of  the  equation, 
the  quadratic  equation  has  the  form 

(5)  ax^  +  hx+c  =  0, 

where  a,  h,  c  are  known  constants.  If  upon  substituting  for  x  a 
particular  number  x^,  the  left  member  of  (5)  reduces  to  zero,  so  that 
tlie  equation  is  satisfied,  x^  is  called  a  root  of  the  equation. 

A  quadratic  equation  (5)  may  often  be  solved  by  inspection  by 
factoring  its  left  member.  Thus  x^  -{-  x  —  12  =  0  may  be  written 
{x  —  ^){x  -[-  ^)  =^  0,  so  that  the  roots  are  +  3  and  —  4. 

When  the  factors  are  not  evident,  we  complete  the  square  of  the 
terms  involving  x.     For  example, 

x^  —  ^x—^l  =  0 
may  be  written,  after  adding  and  subtracting  9, 

(:^;  _  3)'^  _  100  =  0,  {{x  -  ^)  +  10\\{x  -  d)  -  10]  =  0. 
Corresponding  to  the  factors  x-{-7,  :c  —  13,  we  obtain  the  roots 
-7,13. 

31.  To  solve  the  general  equation  ax^  -{-  bx  -\-  c  =  0,  where 
a  9^:  0,  we  divide  by  the  coefficient  a  of  x"^  and  obtain  the  equation 

(6)  xij^^x+-  =  0. 

Adding  and  subtracting  the  square  of  .-  — ,  we  get 

Z  CI 


Kir-£-3=»- 


Since  stny  quantity  is  the  square  of  its  square  root,  the  left  member 
may  be  regarded  as  the  difference  of  two  squares.,  Factoring,  we 
get 

(^+2«  +  ^4a2       aSr^^a       ^ia'      a  f  "  ^-     . 
Each  factor  furnishes  one  of  the  two  roots 


(7)  -=-f±^-=^'. 

^  '  2a  2a 


«EC.  32]  COLLEGE  ALGEBRA.  31 

In  particular,  we  derive  the  factorization 


The  7'oots  of  the  quadratic  eq^iation  ax^  -{-  hx  -\-  c  =  0  are 


—  Z>  +  \/U^  —  4:ac       ^       —  b  —  i/¥  —  4ac 

(9)  a  =:  ^ ,      /3  = '-——^ . 

^  ^  2a  ^  2a 

According  as  P  —  4ac  is  positive,  negative,  or  zero,  the  roots  are  real 
and  unequal,  imaginaj^y  and  unequal,  or  real  and  equal. 
For  the  last  case,  W  —  iac,  formula  (8)  becomes 


ax^  -\-  l)x  -\-  c  =  aix  +        ,  . 

32.  Taking  the  sum  and  the  product  of  the  roots,  we  get 

(10)  a  +  ^=-K      a^  =  tzJ^l^:^)  =  t. 

^     ^  a  4:a^  a 

Comparing  these  results  with  the  quadratic  (6),  we  conclude  that, 
in  a  quadratic  equation  tuhere  the  coefficient  of  x^  is  unity ^  the  sum 
of  the  roots  equals  the  negative  of  tlje  coefficient  of  x^  the  product 
of  the  roots  equals  the  constant  term  (the  term  not  involving  x). 

For  example,  in  the  above  quadratic  x'^  —  Qtx  —  91  =  0,  having 
the  roots  —  -7  and  13,  the  sum  of  the  roots  equals  +  6,  the  product 
of  the  roots  equals  —  91. 

To  give  a  second  proof  by  a  method  applicable  also  to  equations 
of  higher  degrees  (Chapter  XIX),  consider  the  equation* 

(x  -  a){x  -  y^)  =  x'^  —  {oc  +  /3)x  +  a/3  =  0, 

Since  x  —  a  vanishes  for  x  =  a,  the  equation  has  a  root  a.  Simi- 
larly, it  has  a  root  /3.  By  inspection,  the  sum  of  the  roots  is  the 
negative  of  the  coefficient  x-  and  their  product  is  the  constant  term. 

*By  using  the  symbol  =,  read  identically  equal,  we  emphasize  the  fact 
that  we  have  here  two  different  forms  of  the  left  member  of  the  equation,  the 
two  members  of  which  are  connected  by  the  sign  =,  read  equal. 


I 


32  FACTOR    THEOREM;    QUADRATIC  EQUATIONS.       [Ch.  Ill 

The  quadratic  equation  v/hose  roots  are  given  may  therefore  be 
constructed  as  follows: 

x^  —  (sum  of  roots):c  -|-  (product  of  roots)  =  0. 
Thus,  the  quadratic  equation  whose  roots  are  —  2  and  4  is 
^2  _  ( _  2  -f  4).T  -  8  =  a.'2  -  2.^  -  8  =:  0. 

Example  1.  li  a  and  ft  denote  the  roots  oi  x^  —  px  ~{-  q  r=  0^  find  th( 

values  of  ex'  +  fS\    a^  +  fi\    a:^  +  ft\ 

Since  a  -\-  (5  =  jp  and  af5  —  q,  we  have 

^2  +  /52  =:  (or  +  ftY  -  2«:yS  =  f-  -  2g, 

a:3  _^  /53  ^  (tr  4-  fif  -  3aVi  -  3a/?2  ::^  ^3  _  3^^^ 

a*  -f  y^*  =  (a2  4-  ^2)2__  2aVi^  =  (i?^  -"Iqf  -  2^^  ==  p*  _  4^2^  4.  2?^ 

Example  2.  Determine  the  quadratic  equation  whose  roots  are  the  cube< 

of  the  roots  oi  x'  —  px  -^  q  —  ^. 

If  a  and  (i  are  the  roots  of  the  latter,  we  have  (Ex.  1) 

a^  +  /5^  r=  _p3  -  3i?g,     d'ff'  ^  q^. 
The  quadratic  equation  with  the  roots  a^,  fS^  is  therefore 

y'^  -  {p^  -  ^pq)y  4-^  =  0. 
Example  3.  Determine  the  condition  upon  the  coefficients  of  the  equatioi 
ax^  -\-  hx  -\-  c  =  Q  in  order  that  one  root  may  be  four  times  the  other. 

h  c 

By  formulae  (10),  a -{-  /3  = >    aft  =  ~.      Since  ft  is,  by  the  condition 

0/  Ob  ^ 

imposed,  equal  to  4a,  we  must  have 

5a:  = ,     4^2  =  — . 

a  a 

Equating  the  resulting  values  for  d',  we  get  the  condition 

^H'  ^  256^^-. 

EXERCISES. 
Form  the  equations  whose  roots  are 
1.  4,  -  5.  2.  c  4-  (Z,  c  -  d.  3.  0,  4. 

4.  i  -  i.  5.  2  4-  1^3;  2  -  l/a     6.   -  3  4-  1/7;  -  3  -  |/7 

7.  V'^T,  -  \/  '^^.         8.  4  ±  \f  -Z. 
9.  Find  the  condition  that  one  root  of  ax^  -\-  hx  -\-  c  —  ^  shall  be  (1)  th( 
double  of  the  other;  (2)  the  negative  of  the  other. 

10.  If  a  and  ft  are  the  roots  of  x'  —  j?aj  -f  5-  —  0,  find  the  equation  whose  root« 

are  (1)  -^-  and'  -— ;   (2)  a^ft  and  aft'-,  ^3)  p  and  q;  (4)  a  4-  /5  and  a"  4-  , 
(5)  a  4-  I  and  /i+  ^;  (6)  «  +  ^  and  i  4- ^. 


Sec.  33]  COLLEGE  ALGEBRA.  33 

Factor  the  expressions  and  derive  the  roots  of  the  corresponding 
equations: 

11.  x'  -  53a;  +  150  12.  x^  -  ^^x  -  200.  13.  ^x^-  18  -j-  21a;. 

14.  x^  -\-  2ax  —  52  _  2ab.  15.  x'^  —  iax  +  Sa\  16.   x^  —  ^x'^  +  4. 

17.  ^2  _  ^  _  ^2  4.  i         18.   -^  +  ^^-  2.     19.    ^2  +  ^-29 

x^  ■   a^  x  -[-  \  x^  '     ic^ 

33.  Symmetry.  An  expression  is  said  to  be  symmetrical  with 
respect  to  two  or  more  letters  if  they  enter  the  expression  in  such 
a  manner  that  it  is  unaltered  when  any  two  of  the  letters  are  inter- 
changed. 

Thus  2a  +  2Z>  and  a^  ~\-  h^  are  symmetrical  with  respect  to  a  and 
b\  a-\-d  -\-  c  and  ah  -\-  ac  -\-  he  are  symmetrical  with  respect  to  a,  h,  c. 

An  expression  like  ah^  +  ^^  +  ^^^  possesses  only  a  partial  sym- 
metry; it  is  unaltered  when  a  is  replaced  by  h,  h  by  c,  and  c  by  a, 
but  is  altered  if  two  letters  are  interchanged.  It  is  said  to  possess 
cycle-symmetry. 

Example  1 .  Factor  the  cyclo-symmetrical  expression 
;S'  =  a\b  -  c)  +  h\c  —  a)  +  c\a  -  b). 
Writing  b  in  place  of  a,  the  expression  becomes 

b\b  -  c)  +  b\c  -  5)  =  0. 
Hence,  by  the  factor  theorem,  a  —  6  is  a  factor  of  8.     Similarly,  b  —  c  and 
c  —  a  are  factors  of  S,  results  following  also  from  the  cyclo-symmetry  of  S. 
Since  8  is  of  the  fourth  degree  in  the  letters  a,  b,  c,  there  remains  a  fourth 
factor  la-\-mb-^  nc  of  the  first  degree  in  a,  b,  c,  so  that 

o 

(11)  ; — T 77 ^  =  la -\- mb -\~  nc, 

{a  —  b){b  —  c){c  —  a) 

The  numerator  and  denominator  of  the  fraction  both  possess  cyclo-symmetry 
and  therefore  also  the  fraction  does.     Hence 

la  -f-  mb  -\-  nc^lb  -\-  mc  -f-  na. 
From  this  identity  we  derive  I  =  m  =  n.  To  prove  that  ^  =  —  1,  we  observe 
that  the  term  —  a'^(5  —  c)  of  the  denominator  of  (11)  must  divide  into  the  term 
a\b  —  c)  of  8  to  give  the  term  la  of  the  quotient.  Another  method  is  to 
employ  particular  values,  say  a  =  0,  b  =  1,  c  =  2,  when  the  identity  (11)  be- 
comes 

-  6/2  =  m  +  2/i  =  3^. 

Hence  8  =  —  {a  —  b){b  —  c){c  —  a){a  -\-b  -\-  c). 

Example  2.  Factor  the  cyclo-symmetrical  expression 
E~{a-bf^{b-  cf  +  (c  -  af. 


I 


34  FACTOR   THEOREM;    QUADRATIC  EQUATIONS,       [Ch.  Ill 

As  in  Example  1,  we  see  that  E  is  divisible  by  the  product 

P~  [a  -  h){h  -  c){c  -  a). 
Since  E  and  P  possess  cyclo-symmetry,  the  quotient  E/P  is  a  cyclo-symn 
metrical  expression  of  the  second  degree.     Set 

E/P  =  l{a^  -f  b''  +  c2)  -f  m{ab  -[-  5c  -f-  ca). 
For  a  =  Oy  b  =  1,  c  =  —  1,  we  find  that  15  =.  21  —  m.    For  a  =  0,  b—l,  c~  3^, 
we  get  15  =  5^  -j-  2m.     Hence  I  =  6,  m  —  —  ^.     Hence 

E=  5{a—  b){b  —  c){c  —  a){a^  +  b'^  +  c-  —  ab  —  be  —  ca). 

EXERCISES. 
Factor  the  following  cyclo-symmetrical  expressions: 

1.  ab{a  —  b)-{-  bc{b  —  c)  +  ^^^(c  —  a)- 

2.  a\b  -  c)  +  6^{c  -  «)  +  c'(«  ~  *). 

3.  ^25^(61  -  ^>)  H-  b''c\b  -  c)  +  c'-^a2(c  -  a). 

4.  ab{d^  -  b^)  -f  bc{b^  —  c^)  +  ca'c^  ~  a% 
6.  a*(62  -  c')  +  5i(c2  -  a^)  +  c\(i'  -  b''). 
6.  aW  -  c'f  -{-  b\c'  -  a'^f  +  c'^a^  _  52^3^ 


CHAPTER   IV. 

SIMULTANEOUS   EQUATIONS;    DETERMINANTS. 

34.  If  we  assume  that  the  values  of  the  unknown  quantities 
X,  y,  .  .  .  are  the  same  in  each  of  several  equations,  the  equations 
are  called  simultaneous  equations. 

Consider  the  pair  of  simultaneous  equations 

^  ^  ^  6x-2tj  =  2. 

To  solve  for  x,  we  multiply  the  first  equation  by  —  2  and  the  second 
equation  by  -f-  3  and  add  the  resulting  equations.     Then 

14:X  =14,      X  =  1, 

In  a  similar  manner,  we  find  that  y  =  2. 

35.  Consider  the  general  pair  of  simultaneous  equations 
a^x  +  %  =  c^, 

+  hy  =  ^2- 

To  solve  for  x,  we  eliminate  y  between  the  equations.  This  may  be 
done  by  multiplying  the  first  equation  by  b^  and  the  second  by—  b^ 
and  adding  the  resulting  equations.     Then 

{a,b,  -  a,b,)x  =  {c^b,  ~  c^b,). 
Employing  the  respective  multipliers  —  cf^  and  «p  we  get 

The  quantities  in  the  parentheses  are  all  of  the  same  form.  We 
therefore  employ  as  an  abbreviation  the  symbol 

(3)  ^'!;!-"xJ.-«A. 

35 


(2)  \'^' 


36 


SIMULTANEOUS  EQUATIONS;    DETERMINANTS,      [Ch.  IV 


called  a  determinant  of  the  second  order.     It  equals  the  difference  j 
of  the  two  cross-products.     The  above  relations  may  now  be  written  j 


(4) 


X  = 


y 


«o    c. 


By  division,  we  obtain  the  values  x,  y  satisfying  equations   (2).,; 

a,  b. 


In  formulae  (4),  x  and  y  are  each  multiplied  by  the  symbol 


«o  b. 


which  is  formed  of  the  coefficients  in  the  left  members  of  equations: 

c.  b: 


(2).     The  determinant 


c,b. 


is  derived  from  the  preceding  deter- 


y  = 


2-4 

6  2 


minant  by  replacing  the  a's  by  c's.     We  may  express  our  results  by 
the  following  theorem : 

For  the  simultaneous  equations  {2)^x  times  the  determinant  D  of 
the  coefficients  of  the  left  members  equals  the  determinant  obtained  bij\ 
substituting  the  constants  c^^  c^  in  place  of  a^,  a^  in  the  first  column, 
of  D ;  y  times  D  equals  the  determinant  obtained  by  substituting 
the  constants  c^ ,  c^  in  place  of  b^ ,  b^  in  the  second  column  of  D,         ' 
Applying  the  theorem  to  example  (1),  we  get 
2-3      _|- 4     -  31        2-3 
6     -  2  "^  ~|       2     -  2|        6-2 
Evaluating  these  determinants  by  the  definition  (3),  we  get 
14:2;  =  14,     Uy  =  28. 

EXERCISES. 

Solve  by  determinaiits  the  pairs  of  simultaneous  equations  : 
I,  Sx  —  y    =  34,  2,  3aj  +  4y  =  10,  3.     x  —  y    —    5, 

x-\-  Sy  =  5d.  ^x-\-    y  =    9.  5^  -  4y  ==  40. 

4.  ax  —  by  =  c,  5.     x  -}-    y  =  1,  6.  ax  -\-  by  =  a\ 

ex  —  ay  =  b.  ax  -\-  by  =  c.  bx  —  ay  —  ab. 

36.   Consider  the  three  simultaneous  equations 
r  c^i^  +  b,y  +  c^z  =  \ , 
(5)  I  ^\^  +  b.;y  +  c^z  =  \ , 

I  %^  +  b^y  +  c^z  =  \. 


Sec.  36] 


COLLEGE  ALGEBRA, 


37 


ir  we  can  combine  these  equations  in  such  a  way  that  y  and  z 
shall  be  eliminated,  the  resulting  relation  will,  if  it  contains  x,  de- 
termine the  value  of  x.  Multiplying  the  first  equation  by  M^ ,  the 
second  equation  by  M^ ,  and  the  third  equation  by  M^ ,  and  adding 
the  resulting  equations,  we  get 
(0)         {a,M,  +  a,M,  +  a,M^)x  +  {\M,  +  b,M^  +  b,M,)y 

We  desire  tliat  tlie  coefficients  of  y  and  z  shall  vanish.     It  will  be 
shown  that  this  will  indeed  happen  if  we  take 


(7) 


M, 


M,^^ 


K 

c. 

^1                ^1 

1 

1 

,         if3  = 

1                     1 

h 

Cs 

*2         ^'2 

Substituting  for  these  determinants  their  values  as  given  by  the 
definition  (3),  we  find  that 

(8)  h,M^  +  \M^  +  d,M, 

=  ^,(^2^3  -  ^3^2)  -  ^2(^1^3  -  ^3^1)  -r  ^3(^1^2  -  K^i)  ^  0- 
In  a  similar  manner,  c^3f^  +  ^2^2  +  ^3  ^^^3  —  ^-  Hence  (6)  be- 
(!omes 

(9)  {a,  M,  +  a,M\  +  a,M.:)x  =  ^,if,  +  h.M,  +  /^3.¥3. 

For  the  coefficient  of  x,  with  the  values  of  M^ ,  J/^ ,  M^  inserted 
from  (7),  we  shall  employ  the  following  symbol: 
a,     J,     c. 


(10) 


+  ^3 


M 


a,     Z^2     ^2 

«3      ^3      ^3 

It  is  noticed  that  the  symbol  is  formed  of  the  coefficients  of  the 
left  members  of  equations  (5),  the  coefficients  retaining  the  same 
relative  positions  in  the  symbol  as  in  the  equations.  We  may  de- 
rive the  right  member  of  equation  (5)  from  the  coefficient  of  x  by 
replacing  a^,  a^,  a^  by  Jc^ ,  Ic^^h^y  respectively.  Hence  equation  (9) 
may  be  written  in  the  form 


(11) 


«1 

h 

Ci 

K 

K 

c, 

a. 

K 

'-, 

X  = 

K 

h 

^5 

'h 

K 

<\ 

K 

K 

Cr 

■38 


SIMULTANEOUS   EQUATIONS ;    DETERMINANTS.       [Cir.  IV 


Such  symbols  are  called  determinants  of  the  third  order.  If 
the  determinant  on  the  left  is  not  zero,  we  obtain  by  division  the 
value  oi  X. 

37.  The  preceding  method  may  be  seen  to  be  a  direct  general- 
ization of  the  method  employed  in  §35  for  the  solution  of  simul- 
taneous equations  in  two  unknown  quantities.  To  solve  equations 
(2)  for  X,  we  employed  the  multipliers  l^  and  —  h^  [notice  the 
alternation  in  sign^  corresponding  to  the  signs  -{-,  — ,  +  in  the 

^1  w 


multipliers    (7)].      We   observe   that,  in    the  determinant 


V 


^2  stands  diagonally  opposite  to  «j  and  that  ^^  lies  opposite  to  r/./, 
so  that  the  multipliers  used  in  solving  for  x  lie  opposite  to  the  co- 
efficients of  X  in  the  equations  (2).  For  the  three  equations  (5), 
the  nuiltipliers  (7)  used  in  solving  for  x  lie  opposite  to  the  coelli- 
cients  of  x.  Expressed  more  exactly,  the  first  multiplier  M^  niay  ^j 
be  derived  from  the  array  of  coefficients  in  the  left  members  of  (5)*| 
by  erasing  the  row  a^,  h^,  c^  and  the  column  a^,  a^,  a^  containing 

h  c 

of  the  remaininer  coefficients. 


ftj  and  taking  the  determinant 


^3^3 


The  second  multiplier  M^,  apart  from  its  sign^  may  be  derived  by 
erasing  the  row  and  the  column  containing  a^  and  taking  the  deter- 

of  the  remaining  coefficients.     Similarly,  J/gis  gotten 


minant 


^3^3  1 


by  erasing  the  row  and  the  column  containing  a^.     The  resulting 
determinants  are  called  the  minors  of  «i ,  a^^  a^,  respectively. 

In  solving  equations  (2)  for  y,  we  employed  the  multipliers 
—  «2  ^^^  +  ^r  Apart  from  their  signs,  they  are,  respectively,  the 
minors  of  b^  and  h^  (the  coefficients  of  y  in  the  given  equations)  in 


the  determinant 


{h\  for  y,  we  should  employ  as  multipliers 
where 


This  suggests  that,  in  solving  equations 

^1?    +^2'     —   ^3» 


(12) 


-B.^ 


«o 

^, 

0^ 

C^ 

«i 

C,l 

3 

2 

.  ^.= 

I 

1 

.    ~^3  = 

I 

' 

«3 

Cs 

^3 

^3 

fl, 

«^J 

Sec.  38] 


COLLEGE  ALGEBRA. 


39 


which  are,  respectively,  the  minors  of  h^,  1)^,  h^  in  the  determinant 
(10)  of  the  coefficients  of  the  left  members  of  (5).     Then 

(13)  -  a,B,  +  a,B,  -  a^B,  =  0,      -c,B,  +  c,B^  ~  c,B,  =  0, 

so  that,  for  M^=  —  B^,  M^=  +  B^,  M^-  —  B^,  equation  (6) 
gives 

(14)  {~\B,  +  \B,  -  \B,)y  =  -  k,B,  +  \B-\B,. 

The  coefficient  of  y,  with  the  values  of  ^^ ,  B^^  B^  inserted  from 
(12),  may  be  expanded  as  follows: 

(15)  -  \a/.^  +  \a^c^  +  d.a^c^  -  \%c^  -  l^a^c^  +  \a^c^. 

It  is  therefore  equal  to  the  determinant  (10).  The  right  member 
of  (14)  is  derived  from  the  coefficient  of  y  by  replacing  Z>j,  l^,  h^ 
by  k^,  /bj,  ^3,  respectively.     Hence  (14)  becomes 


(16) 


I  Incidentally,  it  was  seen  that  the  determinant  on  the  left  can 
be  expanded  according  to  the  elements  of  the  second  column  by 
multiplying  h^  by  the  negative  of  its  minor  B^^  h^  by  its  minor  B^, 
^3  by  the  negative  of  its  minor  B^^  and  taking  the  sum  of  the  re- 
sulting products. 

Proceeding  in  a  similar  manner,  we  find  that 


«1 

^ 

Ci 

a, 

K 

<■' 

«. 

^ 

c. 

y  = 

a. 

K 

c.^ 

«, 

K 

f. 

», 

K 

c. 

(17) 


«1 

* 

fl, 

T^: 

a. 

I. 

Hence  to  solve  equations  (5)  for  any  one  of  the  unhnoion  quan^ 
titles  Xy  y,  z,  loe  equate  the  product  of  the  unhnoion  and  the  deter- 
minant D  of  the  coefficients  of  the  left  memlers  of  (5)  to  the  deter- 
minant oMained  from  D  hy  substituting  the  constants  k^,  k^^  k^  in 
place  of  the  coefficients  of  that  unknoion. 

38.  It  follows  also  from  the  preceding  developments  that,  if 
A^,  A^,  Jg,  B^,  B^,  B^,  (7i,  Cg,  C^  denote  the  minors  of  «,,  a^,  a^. 


40 


SIMULTANEOUS  EQUATIONS;    DETERMINANTS.      [Ch.  IV 


^1?  ^2^  ^3^  ^1?  ^2'  ^3'  respectively,  in  the  determinant  D,  these  rela- 
tions hold: 

/^i^i-&2^2+M3  =  0,  -b^B^-i-b^B^-b^B^  =  D         b^C^-b^Cr^-^b^C^  -■  0, 


Ci^l-C2^2  +  C3^3    ==0, 


-Ci^i+Cg^g-^sA  =  0>  CjCi-CgCa+c^Ca 


/> 


The  three  expressions  for  D  give  three  methods  of  expanding 
the  determinant  D,  To  expand  D  according  to  the  ^elements  of 
any  column,  we  multiply  each  element  by  its  minor,  with  the 
proper  sign  prefixed,  and  form  the  sum  of  the  products.  The 
proper  sign  depends  upon  the  position  of  the  element,  as  exhibited 
by  the  following  scheme: 


+ 

— 

+ 

— 

+ 

— 

+ 

— 

+ 

Compare  the  black  and  white  spacing  on  a  checker-board. 
39.  The  second  equation  in  the  above  set  may  be  written 

^1     K     ^1 
0  =  b,A,  -  \A^  +  b^A^  =  ^2     K     ^2 

^-^3        h        ^3 

Hence  a  determinant  vanishes  if  its  first  and  second  columns  are 
alike.     Similarly,  the  third  equation  gives 


0 


c,A, 


^2^2      \       ^3^3 


^3         "^3 

Hence  a  determinant  vanishes  if  its  first  and  third  columns  are  alike. 
Similarly,  —  c^B^  -\-  c^B^  —  c^B^  —  0  shows  that  a  determinant 
vanishes  if  its  second  and  third  columns  are  alike.  The  last  result 
also  follows  directly  from  the  definition  (10)  of  the  determinant. 
Combining  these  results,  we  see  that  a  cleterniinant  of  the  third 
order  vanishes  if  any  two  of  its  cohimns  are  alike. 


Sec.  4i)] 


COLLEGE   ALGEBRA, 


41 


40.  Upon  expanding  the  three  determinants 


I 


K 

a. 

<^i 

a.^ 

c. 

h 

c. 

h 

a 

K 

a. 

e. 

> 

a. 

c. 

h 

? 

c^ 

K 

«= 

K 

a. 

c» 

A 

c. 

h 

c. 

K 

a, 

according  to  tlie  elements  l\,  h^^  b^,  we  obtain  in  each  case 

But  these  determinants  were  obtained  from  D  by  interchanging 
two  of  its  columns.  Bi/  the  interchange  of  any  two  columns  of  a 
determinant  of  the  third  order,  its  value  is  changed  in  sign. 

The  theorem  of  §  39  may  be  derived  as  a  corollary  to  the  last 
theorem.  Indeed,  if  D  has  two  columns  alike,  it  is  evidently  un- 
altered upon  interchanging  them,  whereas  it  must  change  in  sign. 
From  D  ~  —  D,  we  derive  D  =  0. 

41.  As  an  example,  consider  the  simultaneous  equations 

^+  y  +  ^  =  11, 

2x  ~  6g  —    z  —    0, 
3:^;  +  4^  +  2z  =    0. 

Employing  the  definition  (10),  we  find  that 

1  1  1 

2  -  6     -  1    =  11. 

3  4  2 
Hence  formulae  (11),  (16),  and  (17)  give 

11      1      1  1     11      1  1      1     11 

tWx^     0-6—1,     lly  =    2       0-1,     11^=2-6       0. 
I  042  302  340 

ach  determinant  has  two  zeros  in  one  of  its  columns.  It  is 
therefore  best  to  expand  the  determinant  according  to  the  elements 
of  that  column.     The  results  are,  respectively, 


1 

-6  -1| 

-11 

3  -1 
3      2 

I( 

mce  X  =  —8,  y  = 

-7,  z 

=  26. 

-11.7,    11 


=  11.26. 


1 


42 


SIMULTANEOUS  EQUATIONS;    DETERMINANTS,       [Ch.  IV 


1  .1    1 

1     1 

1 

a     h     c 

X  — 

h     h 

c 

a^   ¥   c^ 

Jc"    P 

^2 

We    may   factor   D  without 
If  we  substitute  I?  for  a  in  D, 


42.  Consider  the  simultaneous  equations 

^    +  y    +  ^    =h 

(18)  ax  -\-  bi/   -{-  cz   =  Jc, 

c?x  -f-  y^y  -f-  c^z  =  P. 

The  value  of  x  is  given  by  the  relation 


(19) 


Denote  the  coefficient  of  x  by  D. 
making  use  of  its  expanded  form, 
we  obtain  a  determinant  having  its  first  and  second  columns  alike, 
which  therefore  vanishes  by  §  39.  Hence,  by  the  factor  theorem, 
a  —  b  is  Si  factor  of  D.  Similarly,  a  —  c  ernd  b  —  c  are  factors  of 
D,  Since  D  is  of  the  third  degree  in  the  letters  a,  b,  c,  we  may 
write 

(20)  D  =  m{a  -  b){a  -  c){b  -  c), 

where  m  is  a  numerical  factor  not  involving  a,  b,  or  c.  For  the 
particular  values  a  —  0^  b  —  1,  c  =  —  1,  the  identity  (20)  becomes 
2  =  —  2m,  whence  m  =  —  1.  The  same  result  follows  from  a 
comparison  of  the  coefficients  of  a  term,  like  bc'^,  in  the  expanded 
forms  of  the  two  members  of  (20).  Changing  the  sign  of  the 
factor  a  —  c,  we  may  state  the  result: 
1     1     1 

{a  —  b){b  —  c){c  —  a). 


(21)  a     b     c 

d^    W    0^ 

Changing  a  into  k,   we  obtain   (h  —  b)[b  —  c){c  —  k)  as  the 
value  of  the  determinant  in  the  right  member  of  (19).     Hence 

{a  —  b){c  —  a)' 
In  a  similar  manner,  it  is  fouiid  lliat 

{k-  c){a  -Jc)         _  {Ic  -Jt){bj-Jc) 
[b  -  c)[a  -  by     ^  ~  {a"-  a){b  -  r)' 


y 


I 


\ 


Sec.  43]  COLLEGE  ALGEBRA^  43 

43.  If,  in  place  of  the  third  equation  (18),  we  take 
a^x  +  l^y  +  c^z  —  h\ 
the  values  of  x^  y,  z  depend  upon  determinants  of  the  form 

111 
A  ^  a     1)      c 
r/4    ^>4     ^4 

It  follows  as  above  that  A  is  divisible  by  the  product 
P={a-h){b-  c){G-a), 

Since  the  degree  of  A  in  its  expanded  form  is  five,  the  quotient 
A/P  is  of  the  second  degree  in  a,  h,  c.  By  the  interchange  of 
a  and  Z>,  P  is  changed  into  ~  P  and  A  into  —  z/  {§  40),  so  that 
A/P  remains  unaltered.  Likewise  z//P  is  unaltered  by  the  inter- 
change of  a  and  c  or  by  the  interchange  of  h  and  c.  Hence  A/P 
is  symmetrical  in  the  three  letters  a,  J),  c,  so  that  we  may  set 
(22)  A/P  =  m{a?  +  ^^  +  c^)  +  7i{ah  +  ac-\-  he), 

where  m  and  n  are  numerical  values  independent  of  «,  Z>,  c.  We 
may  evaluate  m  and  n  by  substituting  in  the  identity  (22)  special 
values  for  a,  Z>,  c,  such,  however,  that  the  denominator  P  does  not 
vanish.  For  ^  =  0,  b  =  I,  c=  —  Iwe  get  f  =  27)i  —  7i.  For 
a  ~  0,  h  —  1,  c  =  2,  we  get  -^4  =  6m  +  2n.  Solving  these  two 
simultaneous  equations,  we  find  that  m-  =1,  n  =  1.  Hence  the 
above  determinant  may  be  factored  thus : 

^  r={a-  b){I)  -  c){c  -  a){a^  +  V^  -\- c^ -{- ah -\-  ac  +  he), 

EXERCISES. 

Solve  by  determinants  the  following  sets  of  simultaneous  equations,  can- 
celling the  extraneous  factors  in  3,  4,  5,  6  : 

I.     X  -{-    y  -\-    z  =  0,  2.     X  -  2y  -i-    z  =  12, 

oj  +  2^  -f  3s  =  -  1,  ir  +  2y  4-  32!  =:  48, 


aj  +  3y  +  6s  =  0.  6ic  +  4^  +  32  =  84. 

.     x+    y-]-    z  :=!,  ^.    X  -^    y  -\-    z  =  1, 

ax  -\-  by  -\-   cz  =  k,  a^x  +  h'^y  +  c^z  —  W, 

a^x  +  Wy  +  c^z  =  k^.  a^x  -f  Iry  -\-  cH  -  k^, 


44 


SIMULTANEOUS  EQUATIONS;    DETERMINANTS,       [Ch.  IV 


b.  ax  -\-    y  -\-    2  —  a  —  8, 
X  Ar  ay  -\-    z  =  —  2, 
X  -{-    y  -\-  az  —  —  2. 


6.  ax  -\-  by  -\-  cz  =  2a  -[-h  -{•  c^ 
hx  -\-  cy  -{-  az  —  a    +  2&  +  c, 
ex  -\-  ay  -\-  hz  =  a    +  6  -}-  2c. 


44.  If  k^  =  0,  h^  —  0,  h^  =  0,  equations  (5)  become 

(23)  a^x+b^y+c^z  =  0,    a^x-^-b^y-^c^z  =  0,   a^x+b^y+c^z  =  0. 
According  to  the  general  method  of  solution,  we  have 

«j     b^     c, 

(24)  Dx  =  0,     Dy  =  0,     Dz  =  0,     Z>  = 


It  I)  ^  0,  we  obtain  only  the  evident  set  of  solutions  x  =  y  =  z=:0. 
But,  if  Z>  =  0,  the  relations  (24)  impose  no  conditions  upon  x,  y^  %, 
In  this  case,  equations  (23)  might  have  solutions  other  than  the 
evident  set  re  =  ^  =  2;  =  0.  Let  us  seek  the  solutions  for  which 
2;  #:  0,  for  example.  The  first  and  second  equations  (23)  may  then 
be  written 


^1    7.2/ 


•^17-2/ 

^z         ^z 


C.' 


X  u 

For  the  unknown  quantities  —  and  -  we  obtain  the  relations 

z  z- 


(25) 


«1 

h 

X 

-  <^1 

\ 

«1 

\ 

l- 

a. 

-^1 

«2 

h 

z 

-  c. 

\ 

«. 

h 

z 

«2 

-c. 

It  remains  to  inquire  whether  the  resulting  values  of  —  and  — 

z  z 

satisfy  the  third  relation  (23),  which  may  be  written 

(26)  «37+^37  +  '^,  =  0. 

z  z 

^1  ^1 


To  avoid  fractions,  we  multiply  (26)  by 

X      tJ 

the  values  of  — ,  — .     The  result  is  clearly 


a,b. 


—  Cy     b. 

\a. 

—  c, 

a,     b. 

1       1 

7 

+  *, 

1 

+  c. 

1       1 

-c,     b. 

>, 

-c. 

«2  K 

before   substitutinpr 


=  0. 


Sec.  44]  COLLEGE  ALGEBRA.  4i, 

The  left   member   is   seen  to  be  equal  to  D,     Hence,  if  i)  =  0, 

the  third  condition  (26)  is  satisfied  by  the  values  of  — ,   ~  given  by 

z     z 

(25).     We  must  examine  the  special  case  when 


0^  = 


«, 


X      1/ 

vanishes,  since  equations  (25)  then  fail  to  determine  — ,  ~.     If 

z     z 


A= 


^ 


^2         ^2 


does  not  vanish,  we  divide  the  first  and  second  equations  (23)  by  x 
and  obtain  solutions  for  -,  — ,  which  satisfy  the  third  equation  if. 


( 


X       X 

and  only  if,  D  =  0.     Similarly,  if 

a, 


i?,= 


^2 


X       z 
does  not  vanish,  we  obtain  solutions  — ,  — ,  if  Z)  =  0. 

y    y 

There  remains  the  case  A^  =0,  ^3  =  0,  C^  =  0.  Since  a. , 
Z>i,  Cj  are  tacitly  assumed  to  be  not  all  zero,  we  suppose  that  a^  i^  0, 
for  definiteness.  We  may  then  set  a^  =  pa^,  whence  a^h^  —  ap^ 
=:  0  gives  2>2  —  7^^!,  while  a^c^  —  af^  =  0  gives  c^  =  pc^.  Hence  the 
second  equation  (23)  may  be  derived  by  multiplying  the  first  equa- 
tion by  p,  AVe  need  therefore  only  consider  the  first  and  third 
equations  (23).  As  shown  by  the  above  method,  these  two  equa- 
tions determine  the  ratios  of  x,  y,  z  except  in  the  case 


=  0. 


In  the  latter  case,  the  third  equation  is  a  consequence  of  the  first. 
Since  the  equations  then  reduce  to  a  single  one,  arbitrary  values 
may  be  assigned  to  y  and  z;  and  the  equation  then  determines  x. 
Wr  may  therefore  state  the  following  theorem : 


«1 

h 

=  0, 

h 

«i 

=  0, 

«i 

^1 

«» 

h 

^3 

^z 

«s 

Cz 

46 


SIMULTANEOUS  EQUATIONS;    DETERMINANTS.      [Ch.  IV 


The  7iecessary  and  sufficient  conditions  that  equations  (23)  have 
solutions  x^  y,  z^  not  all  zero^  is  that  the  determinant  D  shall  van- 
ish. If  D  —  0,  the  equations  reduce  to  two  equations  or  to  one 
equation,  according  as  the  minors  of  the  nine  coefficients  are  not  all 
zero,  or  are  all  zero.  In  the  first  case,  the  ratios  x  :  y  :  z  are  deter- 
mined., so  that  one  unhnoiim  is  arbitrary j  i7i  the  second  case.,  two  of 
the  unhnowns  may  he  chosen  arbitrarily. 

Given,  for  example,  the  simultaneous  equations 
2^  +  3^  -  4^  =  0,     ^x  +  by-z  =  (),     Ix  +  \ly  —  9^  =  0, 
the  determinant  D  is  zero,  and  relations  (25)  become 


=  17, 


z 


-  10. 


«1 

^ 

c, 

«2 

K 

c. 

a. 

K 

c. 

45.  Theorem.  A  determinant  of  the  third  order  is  not  altered 
in  value  if  tve  multijjly  the  elements  of  any  column  by  a  constant 
and  add  the  results  to  another  cohimn. 

Consider  a  determinant  D  and  its  expansion: 

=:  a^A^  -  a,A,  +  a,A^. 

Multiplying  the  ^'s  by  7n  and  adding  the  products  to  the  a's,  we  get 

a^  +  ^^^1     ^1     ^1 

a^  +  7nb,     b,     c,    =  {a^  +  7nb,)A^  -  {a,  +  7nb,)A^  +  {a^  +  7nb,)A^ 

^3    +    ^^^3         h         ^3 

irr  (a^A^  -  a^A^  +  a,A,)  +  m{b^A^  -  b,A^  +  b,A^)  =  Z>, 
since  the  expression  multiplied  by  m  is  the  expansion  of 

b,     b,     c. 


which  is  zero  by  §  39.  The  theorem  is  therefore  true  for  this  case. 
To  extend  the  proof  to  the  general  case  when  the  elements  of  the 
ith  column  of  D  are  multiplied  by  m  and  added  to  the  /tli  column, 


Sec.  46] 


COLLEGE  ALGEBRA. 


47 


the  resulting  determinant  being  called  Z>„, ,  we  observe  that  the  /th 
column  of  D  may  be  interchanged  with  the  first  column  and  the 
ith  column  interchanged  with  the  second  column  of  the  new  deter- 
minant, giving  the  final  determinant  D^  =  ±  D.  Doing  likewise 
with  D,n ,  we  get  the  determinant  Z>,',  =  ±  D^.  By  the  proof 
given  above,  Z)'„  =  D\  Hence  i>,„  =  D, 
As  an  example,  consider  the  determinant 

2     11 
.     D^     6     2     1 
12     3     1 
Multiplying  the  second  column  by  - 

Oil 

D=  2     2     1 

G     3     1 

Multiplying  tha  tliird  column  by  —  1  and  adding  to  the  second, 


2  and  adding  to  the  first. 


(27) 


0     0     1 
i)  =   2     1     1   =  -  2. 
6     2     1 
The  aim  is  to  obtain  two  zeros  in  one  row  or  one  column. 
46.  Theorem.     If  the  first,  second,  and  third  colnmns  of  a  de- 
terminant of  the  third  order  he  taken  as  the  first,  second^  and  third 
roius,  resijedively^  of  a  new  determinant,  the  resulting  determinant 
equals  the  original  determinant. 

Starting  with  the  determinant  i),  given  by  formula  (10),  we 
form  the  determinant 


i)'  = 


The  terms  of  D  and  D'  which  involve  a^  are  seen  to  be  equal  by 
inspection.  The  terms  of  D*  which  involve  a^  are  —  h^a^c^  -|-  c^afi^^ 
the  same  as  in  B.  The  terms  of  D'  which  involve  a^  are  b^c^a^  — 
^A^3  J  ^1^6  same  as  in  D,     Hence  D  —  D\ 


«1 

«2 

«3 

1^', 

K 

a^ 

a^ 

a^ 

a. 

\ 

K 

^3 

^"l-: 

3 
^3 

-^ 

2 
^2 

3 

+  c. 

h 

c, 

c. 

C, 

48 


SIMULTANEOUS  EQUATIONS;    DETERMINANTS,       [Ch.  IV 


It  follows  also  that  the  determinant  D  maybe  expanded  accord- 
ing to  the  elements  a^,  Z>j ,  6*,    n^  its  first  row,  viz., 

Since  any  row  may  be  interchanged  with  the  first  row  if  we  change 
the  sign  of  D,  we  obtain  the  following  expansions  according  to  the 
elements  of  the  second  or  of  the  third  row : 

D=  -  a^A^  +  \B^  -  c^C^ ,     B  =  a,A^  -  b,B^  +  c,C^. 
For  example,  the  determinant  (27)  may  be  expanded  thus: 


I)  =  0 


0 


+  1 


-2. 


Applying  the  present  theorem  that  the  rows  and  columns  of  a 
determinant  may  be  interchanged  to  the  results  of  §§  39,  40,  wej 
may  conclude  that  the  value  of  a  determinant  of  the  third  order  u 
changed  in  sign  tvhen  any  tiuo  of  its  roios  are  interchanged,  and  also 
that  a  determinani  of  the  third  order  vanishes  if  any  two  of  its\ 
rows  are  alike.  Similarly,  from  the  last  section  we  derive  thd 
theorem  that  a  determinant  of  the  third  order  is  not  altered  in\ 
value  if  lue  multiply  the  elements  of  any  row  hy  a  consta7it  and  add 
the  results  to  the  elements  of  another  row.  ■ 

47.  Theorem.     A  factor  common  to  the  elements  of  any  column 
or  any  row  of  a  determinant  of  the  third  order  may  he  removed  and 
placed  as  a  factor  in  front  of  the  resulting  determinaiit. 
The  proof  follows  from  the  relations : 


ma^ 
ma^ 
mao 


ma 


^2 

mc^ 


^  ma^A^  —  nia^A^  +  ^^^^3^3  =  ^^ 


ma^A^  —  mh^B^  +  mc^C^  =  m 


«. 

\ 

c, 

"2 

K 

c. 

9 

«3 

h 

Cz 

a, 

h 

Ci 

«2 

!>. 

C2 

«s 

h 

Cz 

and  similar  relations  derived  by  interchanging  rows  or  columns. 


Sec.  47] 


COLLEGE  ALGEBRA. 


49 


As  an  application,  consider  the  cyclic  determinant: 
\a     h     c 


A^ 


^3  _|_  J3  _|_   ^3  _   3^^^.^ 


Upon  adding  the  last  two  columns  to  the  first  column,  the  resulting 
determinant,  equal  in  value  to  J  by  §  45,  has  all  the  elements  of  its 
first  column  equal  to  a  -\-  1)  -\-  c,  which  is  tlierefore  a  factor  of  the 
determinant.  Since  aj\-h-^c  and  the  expansion  of  A  are  botli 
symmetrical  in  the  letters  a,  h,  c,  the  remaining  factor  is  a  sym- 
metrical expression  of  the  second  degree.     Hence 

A  =  {a-\-l  +  c)  \m{a?  +  V^  +  c')  +  n{ab  +  ac  +  Ic)']. 
Hence  the  right  member  must  have  unity  as  tlie  coefficient  of  a^ 
ind  zero  as  the  coefficient  of  a%.     Hence  m  =  1,  m  -^n  =  0.     We 
lave  therefore  tlie  important  factorization 
28)     A  =  a^+b^+c^-^abc={a+I)  +  c){a^+b'^+c'^-ab-ac-hc). 

The  factor  of  the  second  degree  may  be  decomposed  into  two 
maginary  linear  factors.  Let  go  be  the  quantity  introduced  in  §  8' 
,0  that  GD^  =  I,  Multiplying  the  elements  of  the  second  column  of 
d  by  00  and  those  of  the  third  column  by  oo^  and  adding  the  prod- 
icts  to  the  elements  of  the  first  column,  the  sums  are 

a  -{-  bao  -\-  CGO^,  c  +  aao  +  boo^  ~   Go{a  -{-  bao  -{-  coo^), 
b  +  coo  -\-  aoo^  =  Go'^{a  +  boo  ~\-  cro^). 
lence  a  +  bGD-\-  coo^  is  a  factor  of  A.     Employing  the  multipliers 
»2  and  G9,  we  obtain  the  factor  a  -{-boo'^  -[-  coo.     Hence 
28')  A  =  {a-[-b-\-  c){a  +  boo -\-  coo^){a  +  boo^  +  coo). 

EXERCISES. 
Solve  the  following  sets  of  simultaneous  equations : 
1.  a;  +   y  +  3z  =  0,        2.     a^  +    3y  +    4^  :=  0,        Z.  2x  -     ^  +    42  =  0, 
aj-f2y+2s=:0,  4a; -I- 12y  4- 162  =  0,  x^    3y  -    22  :=  0, 

a.-f5y-    2=0.  3a!  +    9^  +  122  =  0.  a?  -  lly  +  142  =  0. 

Factor  the  following  determinants  without  expanding  them: 
4. 


a 

b      c 

5. 

1     a    be 

6. 

d 

d 

d 

a" 

52     c' 

1     b    ca 

c 

b 

a 

a^ 

b^    c^ 

1     c    ab 

ab 

ac 

be 

5©  SIMULTANEOUS  EQUATIONS;    DETERMINANTS. 

^  8. 


[Cn.  n 


.    a2    a2  _  (J  _  cf     he  I 

C2        C2     -    (a  -    Z>)2        ,^6  I 

48.  As  definition  of  a  determinant  of  the  fourth  order,  we  take 


d'  h^  6' 

he  ca  ah 


(29) 


^ 


/>. 


a^A^  -  a.^A^  +  a^A^  -  a^A^, 


l^U         ^\         ^4         ^4 

where  .Ij  denotes  the  minor  of  a^  obtained  by  erasing  tlie  row  an^ 
the  cohimn  containing  a^,  A^  the  minor  of  a^,  etc. ;  that  is, 


A^ 


h 

e-i 

d. 

h 

Cz 

d. 

h 

c. 

d. 

\  ^1  ^h 

h  c^  dz 
b,  6\  d. 


A.= 


K 

Ci 

d. 

h 

c^ 

d. 

h 

c. 

d. 

K 

c. 

d. 

h 

^^2 

d. 

K 

^3 

d. 

The  terms  of  (29)  which  involve  5j  are  seen  to  be 


-«A 


c^     d^ 

c^     d^ 

c,     d. 

c,     d. 

+  (^A 

c,     d. 

~  ''*^'  c      d 

=    -b. 

1/3  t,lg 

c,     d. 


c^     d^ 

c,     d. 

c,     d, 

«i     Ci     d, 

^3       3 
c,    d. 

-aA 

c,    d. 

+  a  A 

C3     d. 

=  K 

a,     c,     d, 
««     c,     d. 

which  equals  —  b^B^,  where  B^  denotes  the  minor  of  h^  in  the  deter 
minant  D  defined  by  (29).  Similarly,  the  terms  of  D  which  in 
volve  ?>.,  are 


(^A 


which  equals  ly^B^,  where  B^  denotes  the  minor  of  h^  in  D.  Simi 
larly,  the  terms  of  D  which  involve  h^  may  be  combined  int( 
—  ^3^3,  the  terms  which  involve  h^  may  be  combined  into  h^B^ 
Hence  we  have 

(30)  D=-  \B,  +  \B,  -  \B,  +  ^4^4-  ' 

But  by  interchanging  the  first  and  second  columns  of  D,  we  get 


K 

a. 

Ci 

d, 

K 

". 

^ 

d, 

h 

«» 

Cz 

d, 

K 

a. 

c^ 

d, 

=  \B,  -  \B,  +  \B,  -  h,B, 


D. 


*The  definitions  of  determinants  of  orders  5.  6,  etc.,  are  quite  similar. 
They  have  properties  analogous  to  those  of  determinants  of  orders  2,  3,  4. 


[3ec.  49]  COLLEGE  ALGEBRA,  51 

Hence  a  determinant  of  the  fourtli  order  is  changed  in  sigji  by 
'ntercha7iging  its  first  and  second^  columns. 

Upon  interchanging  the  ^'s  and  c's  in  the  identity  (29),  we  note 
hat  the  signs  of  ^4^,  A^,  A^,  A^  are  changed  (§40).  Hence  the 
lew  determinant  of  the  fourth  order  is  the  negative  of  D,  Simi- 
arly,  upon  interchanging  the  ^'s  and  ^'s,  or  the  ch  and  ^Z's,  the 
ign  of  D  is  changed. 

In  order  to  interchange  the  a's  with  the  c's,  we  may  first  inter- 
ihange  the  a's  with  the  Z>'s,  then  interchange  the  a'^  with  the  c's, 
md  finally  interchange  the  Z?*s  with  the  c's.  The  sign  of  the  de- 
-erminant  is  changed  at  each  of  the  three  steps. 

The  preceding  results  may  be  combined  into  the  theorem: 

A  determinant  of  the  fourtli  order  is  changed  in  sign  ly  inter - 
'hanging  any  tiuo  of  its  columns. 

As  a  corollary  to  this  theorem,  we  have  the  result: 

A  determinant  of  the  fourth  order  vanishes  if  a7iy  two  of  its 
olumns  are  alike. 

49.   By  the  definition  of  the  determinant  symbol,  we  have 
d, 

d^   =  {a^  +  7nb;)A^  -  {a^  +  7nh;)A^ 
^3  +  («3  +  ^'^^3)^3  -  K  +  'inh,)A^ 

=  {a^A^  -  a,A^  +  a,A^  -  a,A^  +  m{\A^  -  \A^  +  h^A^  -  h,A^). 

The  first  quantity  in  parenthesis  equals  the  determinant  (29); 
he  second  equals  m  times  a  similar  determinant  with  the  «^'s  re- 
)laced  by  &'s,  a  determinant  having  its  first  and  second  columns 
like  and  therefore  equal  to  zero  (corollary  of  §48).  We  have 
herefore  proved  that  a  determinant  of  the  fourth  order  is  not 
Itered  in  value  if  the  elements  of  the  second  column  be  multiplied 
•y  a  constant  7n  and  the  results  added  to  the  first  column. 

As  in  §45,  we  extend  the  theorem  to  the  case  in  which  the 
olumns  in  question  are  arbitrary,  say  the  ii\\  and  ji\\  columns, 
ndeed,  by  an  even  number  ofv  interchanges  of  columns,  we  may 


«j  +  mh^ 

\ 

Cl 

flj  +  mh^ 

\ 

«2 

«j  +  mb^ 

\ 

C, 

a^  +  mb^ 

h 

C4 

52 


SIMULTANEOUS  EQUATIONS;    DETERMINANTS,      [Ch.  IV, 


bring  the  iih.  column  into  the  position  of  the  first  column,  and  the 
yth  into  the  second  column.  Each  interchange  of  columns  onlj 
changes  the  sign  of  the  determinant  (§48).  We  may  therefore 
state  the  theorem: 

A  determinant  of  the  fourth  order  is  not  altered  in  value  if  thi 
elements  of  any  column  are  multijMed  hy  a  constant  and  the  ]prod\ 
ucts  added  to  the  corresponding  elements  of  any  other  column. 

As  an  application,  we  prove  that  the  determinant 


J  = 


a^     nc. 


\ 


a^     nc^     c^ 

a^     nc^     c^ 

a^     nc^     c^ 

vanishes  identically.  Multiplying  the  third  column  by  -—  n  and 
adding  to  the  second  column,  we  obtain  an  equal  determinant  hav< 
ing  only  zeros  in  the  second  column.  Interchanging  the  first  and 
second  columns,  we  obtain  a  determinant  which  vanishes,  as  shown 
by  the' definition  (29),  whereas  its  value  is  —  z/. 

60.   Theorem.     A  determinant  of  the  fourth  order  is  U7ialtered 
in  value  if  its  corresponding  roics  and  columns  are  interchanged. 

In  fact,  if  the  given  determinant  is  (29),  the  resulting  deter- 
minant is  i 

«1         «2         ^3         «4 

\    \    h   K 

Cj     c^      c^      c^ 

d^    d^     d^    d^ 

where  A^  is  the  minor  of  a^,  B^  the  minor  of  Z>j,  etc.,  in  D.  In 
view  of  the  corresponding  theorem  for  determinants  of  the  third 
order  (§46),  these  determinants  Jj,  B^,  C^,  D^  are  equal  to  the 
minors  of  a^,  \,  Cj,  d^^  respectively,  in  the  original  determinant 
(29).  Hence  in  the  two  determinants  D  and  (29),  the  terms  in- 
volving a^  are  the  same.  The  terms  of  (29)  which  involve  b^  were 
seen  in  §  48  to  be  given  by  —  l)^B^,  the  same  as  in  D,  The  terms 
of  (29)  which  involve  c^  are  evidently 


D^ 


=  a^A^  -  \B^  +  c^C^ 


dj),. 


F 

Bbc.  51] 


COLLEGE  ALGEBRA, 


{-a,){-c,) 


+  «3(-  ^l) 


-^4(-0 


53 


=    6',  a. 


Similarly,  the  terms  of  (29)  which  involve  d^  are  given  by  —  d^D^ 
It  follows  that  a  determinant  (29)  of  the  fourth  order  can  be 
expanded  according  to  the  elements  of  the  first  row  by  taking  the 
;um  of  the  products  of  each  element  and  its  minor,  with  the  proper 
;ign  prefixed,  viz.,  a^A^  —  h^B^  +  ^i^\  —  dj)^, 
I  61.  Combining  the  theorem  of  §  50  with  the  theorems  of  §§  48, 
p,  we  derive  at  once  the  following  results: 

I  A  determinant  of  the  fourth  order  is  changed  in  sign  hj  inter- 
'hanging  any  two  of  its  roivs.  It  vanishes  if  a7iy  two  of  its  roivs 
•re  alike.  It  is  not  altered  in  value  if  the  elemeiits  of  any  row  are 
mltipUed  ly  a  constant  and  the  products  added  to  the  correspond- 
ng  elements  of  any  other  roiu. 

52.  By  the  definition  (29)  and  the  proof  in  §  50,  a  determinant 
9  of  the  fourth  order  can  be  expanded  according  to  the  elements 
f  tlie  first  column  or  of  the  first  row  as  follows: 
)  =  a^A^  -  a,A,  +  a^A^  -  a^A^,  D  =  a,A^  -  l^B^  +  c^C^  -  dj)^, 
iiiice  any  row  can  be  brought  into  the  first  row  by  an  interchange 
f  rows,  and  likewise  any  column  into  the  first  column,  the  deter- 
linant  being  changed  in  sign  by  each  interchange,  we  derive  the 
xpansions: 

)  ^  -\B^-\-\B-\B,-^lfi„        D  ^  -aJ^+\B,-c,C,+d,D, 
O  =  +  c,C-c,C,+c,C-c,C\,  D  -  -^a,A,-b,B,+c,C-d,D„ 

)  =  -d,D,^dJ)-c\D,-^dJ)„  D=-  aJ,+b,B-c,C,+d,D^. 
'he  sigus  are  determined  by  the  positions  of  the  elements.  The 
olio  wing  scheme,  analogous  to  a  checker-'board,  is  useful: 


+ 

— 

+ 

— 

— 

+ 

— 

+ 

+ 

— 

+ 

— 

— 

+ 

— 

+ 

54 


SIMULTANEOUS  EQUATIONS;    DETERMINANTS.      [Ch.  IV 


53.  As  an  example,  we  evaluate  the  determinant 


D  = 


3  - 

1  - 
-  5  ^ 

2  '  ~ 


I 


We  seek  an  equal  determinant  having  three  zeros  in  one  row  or 
one  column.  To  avoid  fractions,  we  select  a  row  or  column  con- 
taining ±  1  as  an  element,  say  the  last  column.*  Adding  it  to  the 
first  and  third  columns,  and,  after  multiplication  by  7,  to  the  sec- 
ond column,  we  get 

0     -  1 

0 

6  2 

3-10  1-1 


D  = 


0 

-  15 

1 

-2 

0 

-  15 

1 

0 

0 

0 

—  1 

—  

(> 

26 

-  2 

() 

26 

-2 

3 

3 

-  10 

1 

upon  expansion  according  to  the  second  row.  Multiplying  the 
last  row  of  the  determinant  of  the  third  order  by  —  2  and  adding 
to  the  second  row,  we  get 

0 


D=  - 


-  15 

1 

46 

-4 

z=: 

-3 

-  10 

1 

-  15 

46 


=  -  42. 


54.  Consider  the  general  set  of  simultaneous  equations 
a^x  -\-  h^y  +  c^z  -\-  d^w  =p  k^ , 
(^2^  +  \y  +  <^2^  +  d,io  =  h, , 
a^x  +  \y  +  c^z  +  d^io  =  \ , 

To  solve  for  x^  multiply  the  first  equation  by  the  minor  A^,  i\H 
second  equation  hj  —  A^^  the  third  by  A^,  and  the  fourth  by  —  A^ 
and  add  the  resulting  equations.     By  formula  (29),  the  coefficienl 

*  If  none  were  present,  we  drst  work  to  that  end,  unless  all  the  element! 
of  one  row  or  column  are  multiples  of  one  element  fas  is  the  case  with  th( 
first  column  of  the  above  determinant  of  the  third  order). 


Sec.  54] 


COLLEGE  ALGEBRA, 


55 


of  X  in  the  sum  is  the  determinant  D  of  the  coefficients  of  the  left 
members.     The  resulting  equation  becomes 

+  {d^A^-d^A^-\-d^A^-d^A^)w  =  k^A^-h^A^+h^A^-Tc^A^, 

The  coefficients  of  y,  z^  w  all  vanish  since  they  are  the  expan- 
sions of  determinants  having  two  columns  alike  (§48).  The  right 
member  is  the  expansion  of  a  determinant  derived  from  D  upon 
replacing  the  «'s  by  ^'s.     Hence  the  relation  becomes 


Dx^ 


To  solve  for  y^  we  use  the  multipliers  —  ^i ,  -\-  B^,  —  B^, 
-\-  B^,  The  coefficients  of  x,  z^  2V  are  seen  to  vanish,  while  that  of 
y  is,  by  (30), 

-  h,B^  +  b,B,  -  \B,  +  \B,  =  D. 

To  solve  for  z,  we  use  the  multipliers  +  (7^ ,  —  C,,  +  ^3>  —  ^r 
Similarly  for  iv.     We  obtain  the  following  results: 


«1 

K 

C^ 

d. 

K 

\ 

C^ 

(h 

a, 

«3 

C, 

d. 

X  — 

a. 

K 

c. 

ch 

K 

h' 

c. 

d. 

Dy. 


«i  K 

Cj 

d. 

a,   k. 

c. 

d. 

«3      ^3 

^ 

d. 

a.  k. 

c. 

d. 

Dz 


"l 

*i 

h 

< 

a. 

K 

h 

d. 

"3 

h 

K 

d. 

«* 

h 

h 

d. 

Dtv 


«1 

^ 

Cx 

k, 

«2 

K 

<\ 

k 

«3 

h 

Cz 

h 

a. 

I. 

tv 

k, 

Hence  the  product  of  any  one  of  the  unhnown  quantities  hy  the 
determinant  D  equals  the  determinant  obtained  from  D  upon  re- 
placing the  coefficients  of  that  unhnoivn  hy  the  constants  k^,  ,  , .  k^. 

As  an  example  we  take  the  simultaneous  equations 

^-\-  y  +    ^+    w  =  0, 

^+^  +      32;  +      4W  =:  0, 

.^  +  3i/  +    62;  +  lOiu  -  0, 
a;  +  4y  +  IO2;  +  20w  =  —  1. 


56 


SIMULTANEOUS  EQUATIONS;    DETERMINANTS.       [Ch.  IV 


The  determinant  i>  of  the  coefficients  of  the  left  members  is 
found  to  equal  +  1.     Hence 


X  = 


0 

0 

0 

-1 


1 
2 
3 
4 


10    20 


1  1    1 

1  1  1 

1  2 

=  +  1 

2    3      4 

zzz 

0    12 

z:z: 

3    6    10 

0    3    7 

3  7 

1, 


y  = 


1  oil 

1  0     3      4 

1  0      6    10 

1  -1    10   20 


- 1 

111 

1    3      4 

111 
0    2    3 

_      2  3 

1    6    10 

0   5    9 

~       5  9 

Similarly,  z  =  ^^  to  =  —  1,     These  values  of  x,  y,  z,  w  satisfy 
the  given  equations. 

EXERCISES. 

Solve  by  determinants  the  sets  of  simultaneous  equations  : 

2.  ax  -h  by  -\-  cz  -\-  dw  ■-  k, 
a^x  -f  b'^y  4-  c^z  4-  d'^w  =  k\ 
a^x  +  b^y  -\-  c^z  +  d^w  —  J<?y 
a^x  -\-  ¥y  -f  c^z  -\-  d^w  =  k^. 

4.  2x-\-  3y  -  42  -f  hw  =  0, 
3a;  +  5y  -  z  -]-  2w  =  0, 
7aj  +  tly  -  93  +  12^  =  0, 
3a;  +    4y  -  Wz  -\-  13m?  =  0. 


6. 


Solve  by  determinants  the  simultaneous  equations  in  three  unknown 
quantities  to  which  the  following  problems  lead  : 

7.  The  value  of  370  coins  consisting  of  dollars,  dimes,  and  half-dimes 
amounts  to  $61.  If  there  were  twice  as  many  dimes,  half  as  many  half- 
dimes,  and  three  times  as  many  dollars,  the  total  value  would  be  $144. 
Find  the  number  of  coins  of  each  kind. 

8,  How  much  money  shall  be  given  to  A,  B,  C  so  that  A  shall  receive 
one-ninth  as  much  as  B  and  C  together,  B  one-third  as  much  as  A  and  C 
together,  and  C  $6  more  than  A  and  B  together  ? 


1. 

2a?-    y 

-{-dz-2w^ 

-14, 

xJ^^y-\.     z—     W  =  \^, 

3aj  +  5y  -  52  +  3?^  =  11, 

4x-Sy-^2z-    w-  21. 

3. 

6a;  +  4y  +  3z  -  Uw  =  0, 

x  +  2y-j-Sz-4:Sw  =  0, 

a;-2y-{-    z-12w=0, 

4a;  +  4y  -   z  -  24:W  =  0. 

Factor  the  determinants 

6. 

abed 
bade 
e     d    a    b 
d    e     b    a 

0 

a 

b    c 

a 

0 

c     b 

b 

c 

0    a 

e 

b 

a    0 

Sec.  54]  COLLEGE  ALGEBRA.  57 

9.  Find  a  number  such  that  the  sum  of  its  three  digits  is  16,  the  first 
digit  equals  the  sum  of  the  second  and  third  digits,  the  sum  of  the  first  and 
second  digits  equals  seven  times  the  third  digit. 

10.  Find  a  number  the  sum  of  whose  three  digits  is  15,  which  is  increased 
by  99  upon  reversing  its  digits,  and  which  is  decreased  by  9  upon  inter- 
changing the  second  and  third  digits. 


On  the  history  of  determinants,  see  Muir,  Theory  of  Determinants  in  the 
Historical  Order  of  its  Development,  London  (Macmillan  &  Co.),  1890; 
Pascal,  Die  Determinanten,  Leipzig  (Teubner),  1900. 

In  addition  to  the  latter,  the  following  text-books  may  be  consulted: 
Scott,  Theory  of  Determinants,  Cambridge. 
Muir,  Treatise  on  the  Theory  of  Determinants  (Macmillan). 
Weld,  The  Theory  of  Determinants  (Macmillan),  1893. 
Baltzer,  Theode  und  Anwendung  der  Determinanten,  Leipzig. 
Burnside  and  Panton,  Theory  of  Equations,  Dublin  (pp.  229-288). 
Merriman  and  Woodward's  Higher  Mathematics,  New  York  (Wiley 
&  Sons). 


CHAPTER  V. 

RATIO;   PROPORTION;   VARIATION. 

55.  By  the  ratio  of  two  algebraic  numbers  a  and  l  is  meant  the 

quantitative  relation  oi  a  io  h  which  is  measured  by  the  quotient  -. 

A  notation  for  this  ratio  is  «  :  Z>.  By  the  ratio  of  any  two  quan- 
tities of  the  same  kind  is  meant  the  ratio  of  the  two  numbers  ex- 
pressing the  number  of  units  contained  in  the  quantities.     Thus 

fin 
the  ratio  of  2  months  to  10  days  is  expressed  by  the  number  — 

=  6;  the  ratio  of  12  to  20  cents  is  expressed  by  the  number  10. 

Evidently  the  ratio  a  :  h  equals  the  ratio  ma  :  mh.     The  ratio  of 

c  ,     r         ,    c       r       cs 
-,  to  —equals  —  -7-  -  =  -^. 
as  a       s       dr 

If  the  ratio  of  two  quantities  can  be  expressed  as  a  rational 

number,  the  quantities  are  said  to  be  commensurable ;  otherwise, 

they  are  said  to  be  incommensurable.     Thus  the  diagonal  and  side 

i/2 
of  a  square  are  incommensurable,  since  their  ratio  =  J__  =  |/2  is 

not  a  rational  number  (§2). 

The  ratio  aa  :  hft  is  said  to  be  the  ratio  compounded  of  the 

ratios  a  :  l  and  a  :  /?.     Its  value  7-^  equals  the  product  of  -  and 

(X 

-,     When  the  ratio  a  :  &  is  compounded  with. itself  the  resulting 

r' 

58 


Sec.  56]  COLLEGE  ALGEBRA.  59 

ratio  a?  :  IP'  is  called  the  duplicate  ratio  of  a  :  b.     Likewise  a^  :  b^ 
is  called  the  triplicate  ratio  oi  a  \b. 

66.  Problems  involving  several  equal  ratios 
/-.x  ace 

w  i=d  =  j  =  ---  =  ' 

are  usually  solved  most  simply  by  eliminating  the  numerators  a,  Cy 
e,  ...  of  the  fractions  (1),  by  substituting  in  the  proposed  identity 
their  values 
(2)  a  —  rb,     c  =  rd,     e  =  rf,     .... 

Example  1.  Given  —  =  —  =  — ,  prove  that 
b       d       f 

a^  —  2c^  -f  4^^   _  ace 

b^  -  2d''  +  4/3  ~  b^' 

Making  the  substitution  (2),  the  two  fractions  become 

b'  -  2(^  -^  4/3      ~  '^'     ~bdf   ~ 
Example  2.  If  the  ratios  r»  3-'  7*  •  •  •  are  equal,  each  equals 

a  -[-  c  -\-  e  -\-  .  ,  . 


& +  t? +/+... • 
Making  the  substitution  (2),  the  fraction  reduces  to  r 

X       y        z 
Example  3.   Prove  that  -=-=—,  when  it  is  ariven  that 
a       b        c  ^ 

bz  —  cy  _^cx  —  az  _  ay  —  bx 

a       ~~        b       ~        c       ' 

Applying  the  result  of  Ex.  2,  each  of  the  latter  fractions  equals 

a{bz  -  cy)  -\-  b(cx  —  az)  -f  c{ay  —  bx)  _ 

Hence  bz  —  cy  —  0,  ex  —  az  =  0,  ay  —  bx  =  0,  and  therefore 
z  _^  y      X  _z      y  _x 
c  ~  b*     a  ~  c'     b  ~  a' 

EXERCISES. 

1.  Arrange  the  ratios  5:6,    14  :  16,    41  :  48,    31  :  36  in  descending  order 
of  magnitude.     Find  the  ratio  compounded  of  them. 

2.  What  value  of  x  makes  3-|-a;:4  +  «  =  5:6? 

3.  Find  the  duplicate  and  triplicate  ratios  of  3  :  5,  also  of  6  :  7. 


I 


6o  RATIO;    PROPORTION ;    VARIATION.  [Ch.  V 

4.  Find  two  numbers  such  that  their  difference,  their  sum,  and  the  sum 
of  their  squares  are  as  1  :  15  :  113. 

6.  Find  two  numbers  whose  sum  is  77  and  whose  ratio  is  3  :  8. 

It  a  :  b  :=  c  :  d  =  e  :f,  prove  that  ;^| 

6.  a^b''  +  Sa'e^  -  66^f :  b^  +  36^^  _  5;^5  ^  ^4  .  ^4^  .  ' 

7.  a'c  +  Sc^e  +  le^c  :  b^d  +  Sd'e  -f  If'd  =  e^'.p. 

8.  V{ac  -  Zc'  -f  ^e')  :  \/{bd  -  M'  -h  4/*^  =  c  :  c?. 

9.  a2  -f  a6  :  c^  +  c<Z  -  l)^  -  ^ab  :  (Z^  -  6cd. 

10.  If  (a-'  -{-b''  -\-  c2)(aj2  _^  2/2  +  z")  =  (ax  +  by  -\-  cz)\  then  -=  f  =  -. 

a       b        c 

11.  If  ^(wy  +  Tiz  —  ^a?)  =  w(7i2  -\-  Ix  —  my)  =  n{lx  -\-  my  —  nz),  prove 
that  y-\-z  —  x:  I  —  z-\-x— y\m  =  x-\-y  —  z\n. 

12.  If  -jr — ■ — -  — —  ^ — ^-s-,  each  fraction  equals  -. 

^z  -\-  y        z  —  X      5y  —  Sx  ^         y 

13.  If  -^-r  =  r-^  =  — ^,  then  x  +  y -^  z  =  0.  m 

a  —  bb  —  cc  —  a  i^/i  m 

67.  Proportion.  If  the  ratio  a  :  Z>  equals  the  ratio  c  :  d,  the 
four  quantities  «,  h,  c,  d  are  said  to  be  proportionals  or  in  propor- 
tion. The  proportion  is  written  a  :  b  ::  c  :  d,  or  also  a  :  h  —  c  :  d, 
and  is  read  «  is  to  ^  as  c  is  to  d.  The  terms  a  and  d  are  called  the 
extremes,  h  and  c  the  means. 

In  any  proportion  the  product  of  the  extremes  equals  the  product 
of  the  means.     Thus,  from  a  :  b  =  c  :  d  we  get 

-  —  --      ad  =  be, 
b      d 

Inversely,  \i  ad  —  be,  then  a,  b^  c,  d  are  in  proportion. 

If  three  quantities  a,  b,  c  are  such  that  a  \  b  =■  b  \  c^  so  that 
J2  —  ac,  then  b  is  said  to  be  a  mean  proportional  between  a  and  c, 
while  c  is  said  to  be  a  third  proportional  to  a  and  b. 

If  a  \  b  =  c  \  d,  then  a  ±  b  :  b  =  c  ±  d  :  d.     In  proof,  we  add 

±  1  to  each  term  ot  v  =  -i-     Then  — 7—  =  — - — . 
b      d  b  d 

Dividing  the  terms  of  the  equation  in  which  the  plus  sign  holds 

by  the  terms  of  the  equation  in  which  the  minus  sign  holds,  we  get 

a'\-  b  _c  -\-  d 

a  —  b^  c  —  d' 


Sec.  58]  COLLEGE  ALGEBRA,  6i 

Hence,  ii  a  :  b  =  c  :  d,  then  a  -{-  b  :  a  —  b  —  c~{-d:c  —  d,a,  result 
said  to  be  derived  by  composition  and  division  from  a  :  b  =  c  :  d. 

EXERCISES. 

1.  It  a:  b  =  b  :  c,  then  a  :  c  =  a^  :  b'K 

2.  If  a  :  b  =  c  :  d,  then  b  :  a  =  d  :  c,  and  a  :  c  =  b  :  d. 

3.  It  a  :  b  =  c  :  d,  and  e  -.f  =  g  :  h,  then  ae  :  bf  =  eg  :  dh. 

4.  Find  the  mean  and  third  proportionals  to  4  and  16;  to  a*  and  a^U*. 

It  a  '.  b  ■=  c  :  d,  prove  that 

5.  a'c  -  ac'  :  bH  -  bd''  =  {a  -  cf  :  (b  -  df. 

6.  pa^  -f-  qh'^  :  pd^  —  qb'^  =  pc^  +  9'^^  :  pc^  —  qd^, 
1.  b  -  d:a  -  c=  \/l}'  4-  d'  :    i^a'-^  +  cK 

It  a  :  b  =  b  :  c  =  c  :  d,  prove  that 

8.  a:b-\-  d^  c^  -.c'd-i-  d\ 

9.  a  +  5c?  :  2(^  -  3(^  =  a^  _[_  5^3  .  2a^  -  Sb\ 

10.  id'  +  6^  +  c2)(62  -f  c^  +  (?2^  :=  (a6  -{-  bc-h  edf. 

11.  Find  four  proportionals  the  sum  of  whose  squares  is  530,  the  sum  of 
the  extremes  being  23  and  the  sum  of  the  means  13. 

58.  Variation.  If  two  variable  quantities  A  and  B  depend  upon 
each  other  in  such  a  way  that  when  A  is  changed  in  a  certain  ratio 
B  is  changed  in  the  same  ratio,  B  is  said  to  vary  directly  as  A . 
For  example^  a  body  moving  at  the  uniform  rate  of  5  miles  an  hour 
will  move  10  miles  in  2  hours,  30  miles  in  6  hours,  etc.,  so  that 
the  distance  varies  as  the  time.  Between  the  distance  B  reckoned 
in  miles  and  the  time  A  reckoned  in  hours,  the  following  relation 
holds:  B  —  6 A,  In  the  general  case,  if  B  varies  directly  as  ^, 
then  B  =  niA,  where  m  is  some  constant  number. 

If  B  varies  directly  as  the  reciprocal  (or  inverse)  of  A,  then  B 

Q 

is  said  to  vary  inversely  as  A^  and  B  =  -j,  where  c  is  some  con- 
stant number. 

69.  The  Law  of  Boyle  states  that  the  volume  of  a  given  mass  of 
any  gas  at  a  constant  temperature  varies  inversely  as  the  pressure. 
Thus  if  V  is  the  volume  when  the  pressure  is  F,  the  volume  be- 
comes ^Fwhen  the  pressure  is  3P,  the  volume  becomes  ^^Fwhen 

the  pressure  is  —P. 


I 

62  RATIO;    PROPORTION;    VARIATION.  [Ch.  V 

The  Law  of  Gay-Lussac  and  Charles  states  that,  if  the  pressure 
be  constant,  the  volume  of  a  mass  of  gas  varies  directly  as  the  tem- 
perature measured  in  degrees  Centigrade*  above  —  273°  C.  A  gas 
therefore  expands  by  -^^  of  its  volume  at  0°  for  every  'increase  in 
temperature  of  l""  Centigrade.  Thus  273  volumes  of  air  at  0° 
become  273  +  ^  volumes  at  f  Centigrade,  and  the  latter  become 
273  +  T  volumes  at  1  °. 

From  the  preceding  law  and  the  Law  of  Boyle,  we  would  expect 
that,  when  the  pressure  P  and  the  temperature  7'(in  degrees  above 

T 

—  273°  C.)  both  vary,  the  volume  Fof  the  gas  would  vary  as  ^, 

By  actual  experiment  this  statement  is  found  to  be  true  for  mod- 
erate values  of  Tand  P.  It  may  however  be  derived  from  the  two 
laws  stated  by  means  of  the  following  algebraic  theorem : 

60.  If  A  depends  only  on  B  and  C,  and  if  A  varies  as  B  iclien 
C  is  constant,  and  if  A  varies  as  G  tvJien  B  is  constant^  then  A  will 
vary  as  the  product  BC  when  B  and  G  change  simtiltaneously . 

Let  the  quantities  A,  B,  Chave  initially  the  values  a,  h,  c  and 
suppose  that  a  change  in  B  from  Z>  to  ^  and  a  simtdtaneotis  cliange 
in  G  from  c  to  y  together  cause  a  change  in  A  from  a  to  a.  We 
may,  however,  consider  that  the  changes  in  B  and  G  take  place 
successively.  First,  if  G  remains  at  the  constant  value  c  while  B . 
changes  from  h  to  /?,  then  A  will  change  from  a  to  some  value  «' 
intermediate  to  a  and  a.  Applying  the  hypothesis  for  this  case, 
we  get 

a  _  b 
a'^p' 

Next,  let  B  remain  at  the  constant  value  /?,  which  it  just  attained, 
while   G  now  changes  from  c  to  y.      Then  A  must  complete  its 


*  The  freezing-point  of  pure  water  is  33°  Fahrenheit  or  0°  Centigrade;  the 
boiling-point  of  water  is  212°  Fahrenheit  or  100°  Centigrade.  Hence  180  de- 
grees Fahr.  =  100  degrees  Cent. 


Sec.  60]  COLLEGE  ALGEBRA.  63 

change  from  the  intermediate  value  a'  to  the  final  value  a.     Ap- 
plying the  hypothesis  for  such  a  change,  we  get 


a        y 

Multiplying  together  the  two  equations,  we  get 

a  _   be 
a  ~  py' 

Hence,  if  B  and  (7  change  simultaneously,  A  varies  as  BC. 

Example. — Triangles  with  the  same  base  are  proportional  to  their  alti- 
tudes, and  triangles  witli  the  same  altitiide  are  proportional  to  their  bases. 
Hence  the  areas  of  triangles  vary  as  the  products  of  their  bases  and  altituiles. 

EXERCISES. 

1.  If  X  varies  as  y  and  if  a;  =  6  when  y  —  15,  find  x  when  y  =  10. 

2.  If  X  and  y  each  vary  as  z,  then  x  -{-  y  and  \/xy  each  vary  as  z. 

3.  If  y  varies  as  a  -f-  b,  and  a  varies  directly  as  x,  and  h  varies  inverse  1}^ 
as  x"^,  and  if  ?/  =  19  when  ic  =  2  or  3,  find  y  in  terms  of  x. 

4.  If  X  varies  directly  as  y  and  inversely  as  z,  and  \i  x  —  2  when  y  —  o 
and  z  —  4:,  find  y  when  x  —  12  and  2  =  6. 

5.  When  a  body  falls  from  rest  its  distance  from  the  initial  point  varies 
as  the  square  of  the  time  it  has  been  falling,  and  its  velocity  varies  as  the 
time.  If  a  body  falls  400  feet  from  rest  in  5  seconds,  how  far  does  it  fall  in 
10  seconds?  How  far  in  the  tenth  second?  If  the  velocity  at  the  end  of  two 
seconds  is  64,  what  is  the  velocity  at  the  end  of  5  seconds  ?  At  the  end  of  10 
seconds  ? 

6.  An  amount  of  gas  measures  100  cubic  feet  at  0°;  find  its  volume  at  10° 
Centigrade,  the  pressure  remaining  constant. 

7.  Given  500  cubic  feet  of  air  at  10°  Centigrade,  find  its  volume  at  —  10° 
C,  the  pressure  being  unchanged. 

8.  If  the  temperature  of  a  gas  is  raised  from  0°  to  30°  Centigrade,  and  the 
pressure  increased  .tenfold,  what  becomes  of  500  cu.  ft.  of  air  ? 

9.  What  is  the  increase  in  pressure  of  the  air  in  an  air-tight  room,  when 
the  temperature  is  raised  from  0°  to  40°  Centigrade? 

10.  Given  that  the  area  of  a  circle  varies  as  the  square  of  its  radius,  show 
that  a  circle  of  5  inches  radius  equals  the  sum  of  two  circles  of  radii  3  and  4 
inches. 


CHAPTER  VI. 

ARITHMETICAL,    GEOMETRICAL,    AND   HARMONICAL 
PROGRESSIONS. 

61.  A  series  of  quantities  is  said  to  be  in  arithmetical  progres- 
sion  when  the  difference  between  any  term  (after  the  first)  and  the 
preceding  term  is  the  same  throughout  the  series. 

The  following  series  are  examples  of  arithmetical  progressions:; 

2,  4,  6,  8,  .  .  .,  271,  .  .  . 

-1,        -3,  ~5,  -7,   .  .  .,        -  (2;i-l),  .  .  . 

a,    a-{-  d,    a  +  2d,    a  -\-  3d,  .  ,  .,    a  +  {n  —  l)d,  .  .  . 

In  the  first  series  the  common  difference  is  2,  in  the  second 
series  it  is  —  2,  in  the  third  series  it  is  d.  The  nth  term  in  the 
first  series  is  2n,  in  the  second  series  —  {2n  —  1),  in  the  third  series 
a-{-  {n  —  l)d.  In  particular,  for  n  =  4,  the  fourth  terms  in  the 
respective  series  are  2-4  =  8,  —  (2-4  —  1)  =  —  7,  «  +  (4  —  1)^^ 
z=z  a  -\-  3d.     We  may  state  the  theorem : 

In  an  arithmetical  progression  tvith  the  first  term  a  and  the 
common  difference  d,  the  nth  term  is  a  -|-  {n  —  \)d, 

62.  When  three  quantities  are  in  arithmetical  progression,  the 
middle  one  is  called  the  arithmetical  mean  of  the  other  two. 

But,  if  a,  Z>,  c  are  in  A.  P.*,  we  have 

h  —  a  =  c  —  b  =  common  difference. 

*A.  P.  is  an  abbreviation  for  arithmetical  progression.  Similarly,  G.  P. 
will  be  used  as  an  abbreviation  for  geometrical  progression,  H.  P.  for  har^ 
monical  progression. 

64 


Sec.  63]  COLLEGE  ALGEBRA,  65 

Heuce  h  =  |(«  +  c),  so  that  tlie  arithmetical  inean  of  any  two  quan- 
tities is  half  their  sum.     It  is  their  ''  average  value." 

When  any  number  of  quantities  are  in  arithmetical  progression, 
the  terms  between  the  first  and  last  are  called  arithmetical  means 
between  the  first  and  last  quantities. 

Thus  7,  10,  13,  16,  19  are  in  A.  P.  and  10,  13,  16  are  the  three 
arithmetical  means  between  7  and  19. 

63.  Sum  of  a7iy  7iumher  of  terms  in  arithjnetical  progression. 

Let  S  denote  the  sum  of  n  terms  in  A.  P.  of  which  a  is  the  first 
term,  I  the  last  (viz.,  the  ni\\)  term,  and  d  the  common  difference. 
By  §  61,  Z  =  rt  +  {n  -  l)d.     We  have 

S=a+{a  +  d)  +  {a+  2d)  -\-  .  .  ,  J^  {I  -  2d)  +  {I  -  d)  +  I 

Writing  the  series  in  reverse  order,  we  get 

S=l+{l-d)  +  {l-2d)  +  ,..  +  {a  +  2d)  +  {a  +  d)  +  a. 

Adding  the  corresponding  terms  of  the  two  series,  we  get 

2S={a  +  l)  +  {a  +  l)  +  {a  +  l)  +  ,..  +  {a  +  l)=n{a  +  l), 

[there  being  n  terms  each  {a  -\- 1).     Hence 

'(1)  8=M<^  +  1). 

Here  ^{a  +  I)  is  the  average  of  the  first  and  last  terms  of  the 
series,  also  the  average  of  the  second  term  and  the  term  preceding 
the  last,  etc.  If  71  is  odd,  there  is  a  middle  term,  the  ^{n  +  l)st 
term  of  the  series,  whose  value  is  seen  to  be 

a  +  {^n  +  1)  -  1  \d  =  !{«  +  a  +  {n-  l)d\  =  ^a  +  0- 

Hence  ^{a  +  I)  is  the  average  throughout  the  series.     We  derive  a 

I  7 

second  proof  that  the  sum  of  n  terms  is  n    7"   . 

64.  Example  1.  Find  the  sum  of  19  terms  of  the  A.  P.  1,  5,  9,  .  .  . 
The  19th  term  Hs  1  +  (19  -  1)4  =  73.     The  required  sum  is 

19  .  ^-~  =  19  .  37  -  703. 


66  ARITHMETICAL  PROGRESSION.  [Ch.  VI 

Example  2.  The  11th  term  of  an  A.  P.  is  12,  and  the  19th  term  is  36;  find 
the  40th  term. 

The  conditions  give 

a-{-10d  =  12,     a+lSd=  36. 
Hence  8^^  =24,     d  =  3,  whence  a  =  —  IS.     Then  the  40th  term  is 
a  -f  396?  =  -  18  +  39  .  3  =  99. 

Example  3.  Insert  six  arithmetical  means  between  8  and  29. 

We  are  to  construct  an  A. P.  of  8  terms  such  that  the  first  term  a  is  8  and 
the  eighth  term  a  -{-  Idis  29.      Hence  Id  =  21,  d  =  d.     The  required  means  \ 
are  therefore  a  +  d  =  11,  a  ^  2d  =  U,  17,  20,  23,  26.  ' 

Example  4.  How  many  terms  of  the  A.  P.  48,  40,  32,  .  .  .  must  be  taken  i 
so  that  the  sum  may  be  144? 

Let  the  number  of  terms  be  n.  Since  the  common  difference  is  —  8,  the  i 
nib.  term  is  ?  E  48  —  S{n  —  1).     Hence 

144  =  /S  =  ^n{a  -{-  I)  =  ln{9^  -  S{n  -  1)]. 
Hence  144  =  n{62  —  4/1),  so  that 

7i2  —  ISn  -\-2Q~in  -  4)(/i  -  9)  =  0. 

For  71  =  9,  we  get  the  A.  P.  48,  40,  32,  24,  16,  8,  0,  -  8,  -  16,  whose  sum  is 
144.     Since  the  last  five  terms  have  the  sum  zero,  7i  =  4  is  a  suitable  value. 

EXERCISES. 

1.  Find  the  13th  and  41st  terms  of  the  series  6,  12,  18,  .  .  . 

2.  Find  the  20th  and  40th  terms  of  the  series  —  5,  —  3,  —  1,  .  .  . 

3.  Find  the  10th  and  60th  terms  of  the  series  1,  6,  11,  .  .  . 

Find  the  sum  of  the  following  series  : 

4.  5,  11,  17,  .  .  .  to  30  terms.  5.  12,  9,  6,  ...  to  21  terms. 
6.  2^,  4,  5|,  ...  to  37  terms.  7.  a,  Sa,  oa,  ...  to  a  terms. 
8.  a,  0,  —  a,  .  .  .  to  4:a  terms.              9.  13,  9,  5,  ...  to  100  terms. 

Find  the  common  difference  and  the  number  of  terms  in  the  A.  P. ; 

10.  The  first  term  is  6,  the  last  term  180,  the  sum  2790. 

11.  The  sum  is  72,  the  first  term  27,  the  last  term  —  18. 

12.  The  last  term  is  —  32,  the  sum  —  266,  the  first  term  -  6. 

13.  The  first  term  is  a,  the  last  term  ISa,  the  sum  49a. 

Find  the  A.  P.  in  which 

14.  The  7th  term  is  1  and  the  31st  term  is  —  77. 

15.  The  12th  term  is  214  and  the  41st  term  is  739. 

16.  The  54th  term  is  —  125  and  the  4th  term  is  zero. 

How  many  terms  must  be  taken  of 

17.  The  series  15,  12,  9,  ...  to  make  the  sum  45? 

18.  The  series  42,  39,  36,  .  .  .  to  make  the  .  um  315? 


Sec.  65]  COLLEGE  ALGEBRA.  67 

19.  The  series  16,  15,  14,  ...  to  make  the  sum  100? 

20.  The  series  -  10|-,  -  9,  -  7i  to  make  the  sum  —  42? 

21.  Insert  7  arithmetical  means  between  269  and  295. 

22.  Insert  15  arithmetical  means  between  23  and  71. 

23.  Insert  8  arithmetical  means  between  —  80  and  —  50. 

24.  Insert  10  arithmetical  means  between  6^  —  5^  and  Qy  —  5aj. 

25.  Find  5  numbers  in  A.  P.  whose  sum  is  80  and  the  sum  of  whose 
I  squares  is  1640. 

I       26.  Find  three  numbers  in  A.P.  whose  sum  is  39  and  product  2184. 

27.  The  sum  of  the  first  n  odd  numbers  is  n\ 

28.  Find  the  sum  of  all  the  odd  numbers  between  200  and  400. 

65.  A  series  of  quantities  is  said  to  be  in  geometrical  progres- 
sion when  the  ratio  of  any  term  (after  the  first)  to  the  preceding 
term  is  the  same  throughout  the  series. 

The  following  series  are  examples  of  geometrical  progressions: 

1,       2  ,       4    ,      8    ,      .  .  .,       2«-i  ,      2-    .   ,      ... 

q  Q  -1  1  (l\n-i  (\\n-2 

(a,  ar,  ar^,  ar^,  .  .  ,,  ar''~'^,  ar""  ,  ... 
In  the  first  series  the  commoii  ratio  is  2,  in  the  second  series  it 
\,  in  the  third  series  it  is  r.  The  ^th  term  in  the  first  series  is 
-\  in  the  second  series  ^{\Y~^,  in  the  third  series  ar""-^.  In 
particular,  for  n  —  4,  the  fourth  terms  in  the  respective  series  are 
h2^  =  8,  9{|-)^  =  i,  ar^.  We  may  state  the  theorem: 
m ://?'  a  geometrical  jjrogression  with  the  first  term  a  and  the  com- 
IB^  ratio  r,  the  nth  term  is  ar'^~'^. 

^  66.  When  three  quantities  are  in  geometi'ical  progression,  the 
middle  one  is  called  the  geometrical  mean  of  the  other  two. 
But,  if  «,  h,  c  are  in  G.P.,  we  have 

he 

~  =  Y  =  common  ratio. 

a       b 

Hence  h'^  =  ac,  so  that  the  geometrical  mean  of  any  two  qxiantities 
is  tlie  square  root  of  their  product.  It  follows  from  §57  that  the 
•geometrical  mean  of  a  and  c  is  the  mean  proportional  between 
a  and  c. 


6S  GEOMETRICAL  PROGRESSION.  [Ch.  VI 

When  any  number  of  quantities  are  in  geometrical  progression, 
the  terms  between  the  first  and  last  are  called  geometrical  means 
between  the  first  and  last  quantities. 

Thus  2,  10,  50,  250,  1250  are  in  G.  P.,  and  10,  50,  250  are  the 
three  geometrical  means  between  2  and  1250. 

67.  Theorem.  The  sum  of  n  terms  of  a  geometrical  progression 
whose  first  term  is  a  and  common  ratio  r  is 

^      Since  the  ^th  term  is  ar'^~'^,  we  have 
4?^  =  a  +  ar  +  «r2  +  .  .  .  +  ar""-^  =  a(l  +  r  +  r^  +  . . .  +  r''-^)A 
But  formula  (1)  of  §  28  gives 

r^  —  1      1  —  r'* 


14-r  +  r2  +  .  .  .  +r~- 


r  —  1        1  — r' 


68.  As  an  example,  the  sum  of  20  terms  of  the  geometrical 
progression  1,  |^,  ^,  .  .  .  is 


S. 


1  —  /iASO 


The  sum  of  50  terms  is  2  -  (|)*^  the  sum  of  100  terms  is  2  —{^y\ 
It  appears  that  the  sum  of  n  terms  8^  is  less  than,  2,  but  differs  from 
2  by  an  amount  which  decreases  as  n  increases.  In  fact,  S^^ ,  S^ , 
8,^  differ  from  2  by  (^y\  (i)^  (|)^  respectively.  By  taking  n 
sufficiently  large,  8^  will  differ  from  2  by  as  little  as  we  please.  In 
fact,  for  each  increase  of  n  by  unity,  the  difference  in  question  is 
divided  by  2.  The  statement  is  made  more  evident  by  observing 
that  {\Y  =  yV  <  TTT'  so  that 

/1\*"*         1  I 

\2 )      <  10^  ^  •  ^^  •  •  •  ^M^  -  1  zeros  before  1).  1 

Hence  (|)*"*  may  be,  made  as  small  as  we  please  by  taking  m  large 
enough.     We  have  established  the  following  theorem: 


Sec.  69]  COLLEGE  ALGEBRA.  69 

By  taking  a  sufficiently  large  numier  of  terms,  the  sum 

(3)  l  +  i  +  i  +  i  +  .-. 

can  he  made  to  differ  from  2  hy  as  little  as  we  please. 
The  theorem  may  be  iUustrated  by  the  diagram : 

I i 1    ^    1^  1^1  ■ 

A  line  two  inches  long  is  divided  into  two  equal  parts;  the  second 
part  is  subdivided  into  two  parts  each  ^  inch  long;  the  second  sub- 
part is  divided  into  two  parts  each  \  inch  long,  etc.  Taking  the 
first  of  each  pair  of  parts  and  forming  the  sum,  we  get  the  expres- 
sion (3).  After  any  number  of  steps,  the  sum  is  less  than  the 
length  2  inches  of  the  whole  line;  but  the  difference  from  2  inches, 
being  the  last  piece  of  the  line,  tends  towards  zero  as  the  operation 
is  continued. 

69.  For  the  general  series  a,  ar'^,  ar^,  .  .  .,  the  sum  S,^  differs 

a                                       ar^ 
from  - — '■ —  by  the  quantity .     If   r  is   a  proper  fraction, 

whether  positive  or  negative,  the  numerical  value  of  r^  decreases  as 
n  increases  and  may  be  made  to  approach  zero  as  near  as  we  please 
by  sufficiently  increasing  the  value  of  71.  We  will  denote  by  S^ 
the  limit  of  the  sum  Sn  of  n  terms  when  n  increases  without  bound, 
and  speak  of  the  result  as  the  sum  to  infinity.  Hence,  in  a  geo- 
metrical progression  with  the  first  term  a  and  common  ratio  r  nU' 
merically  <  1,  the  sum  to  infinity  is 

(4)  S^ 


1  -  r 

Notice  that,  in  a  geometrical  progression  with  r  >  1,  the  result 
(4)  no  longer  holds.     For  example,  if  a  =  1,  r  =  2,  the  sum 

1  +  2  +  4  +  8  +  16  +  .. .  +  2^-^ 

increases  without  bound  when  the  number  of  terms  n  increases, 

whereas has  the  value  —  1. 


70  GEOMETRICAL  PROGRESSION  [Ch.  VI 

Example  1.  Find  the  geometrical  progression  whvose  second  term  is  —  6 
and  whose  sum  to  infinity  is  8. 

Let  a  be  tiie  fir^t  term  and  r  the  common  ratio.     Then 

ar  =  -  6,     ---^  =z  8. 
1  -  r 

By  division,  r(l  —  r)  =  —  ~ , 

Solving  this  quadratic  equation,  r  =  -  |  or  +  |.  The  value  r  =  |  is 
impossible,  since  8^  would  then  exceed  8.  Hence  r  =  —  |,  so  that  the 
series  is  12,  —  6,  3,  .  .  . 

Example  2.  Find  the  value  of  the  recurring  decimal 

.6  23  E. 6  23  23  23  .  .  . 
Aside  from  .6,  it  may  be  regarded  as  a  G.  P.  with  the  ratio  (x\j)'s 
6         2^    .    _^    ,    ^^    . 


""lo'^wv  ^102  ^  10*  "^  •  •  7 

_  6         23  /      1      \   _  ^     ,   ^3    100 
"  To  "^  lO'i  1  _  _L  )  ~  10  "^  W '  99 
\        lOV 

_  6^        23   _  617 
"■  10  "^  990  "  990* 

EXERCISES. 

1  Sum  12  -[-  18  +  27  +  .  .  .  to  8  terms,  also  to  n  terms. 

2  Sum  36  —  12  +  4  —  ...  to  7  terms,  also  to  n  terms. 

3  Sum  f^       2  4-  I  —  •  •  •  to  71  terms   also  to  infinity. 

4  i  uiu  .5  -f    15  -f  -045  -{-...  ton  terms,  also  to  infinity. 
6  Evaluate  .023;  1.466;  .142857;  .052. 

G.  Find  the  G.  P.  \utli  2d  term  —  4  and  sum  to  infinity  9. 

7.  Find  three  numbers  in  G.  P.  such  that  their  sum  is  62,  and  the  sum  of 
their  squares  is  2604 

8.  If  an  odd  number  of  quantities  are  in  G.  P.,  show  that  the  first,  the 
middle,  and  the  last  of  them  are  also  in  G.  P. 

9.  Insert  between  4  and  18  two  numbers  such  that  the  first  three  shall  be 
in  A.  P.  and  the  last  three  in  G.  P. 


Sec.  70]  COLLEGE  ALGEBRA.  71 

10.  Find  three  numbers  in  G.  P.  whose  product  is  1728  and  the  sum  of 
whose  products  by  twos  is  624. 

11.  Insert  four  geometrical  means  between  5  and  IGO  ;  between  \  and  27. 

12.  Find  a  G.  P.  with  81  as  the  5th  term  and  21  as  the  2d  term. 

13.  Find  a  G.  P.  the  sum  of  whose  first  and  second  terms  is  5  and  such 
that  every  term  is  3  times  the  sum  to  infinity  of  all  the  terms  that  follow  it. 

14.  Sum  to  n  terms  x  -\-  a,  x^  -\-  2a,  x^  -j-  3a,  .  .  . 

15.  Sum  to  2n  terras  (t  -|-  3,   'Sa  —  6,   5a  -{-  12,  .  .  . 

16.  Find  the  ratio  of  two  numbers  whose  arithmetical  mean  is  double  the 
geometrical  mean. 

17.  Find  a,  b,  c,  given  that  their  sum  h  70,  that  a,  b,  c  are  in  G.  P.,  and 
that  Aa,  5\  4c  are  in  A.  P. 

18.  Explain  the  paradox  of  the  race  between  the  hare  and  the  tortoise. 
The  latter  travels  at  the  rate  of  1  mile  an  hour  and  the  former  travels  2  miles 
an  hour.  The  tortoise  has  a  start  of  one  hour.  While  the  hare  is  covering 
the  distance  the  tortoise  travelled  during  the  preceding  period,  the  tortoise 
moves  ahead  a  new  distance,  etc.     Does  the  hare  overtake  the  tortoise  ? 

70.  Several  quantities  are  said  to  be  in  harmonical  progres- 
sion when  their  reciprocals  are  in  arithmetical  progression.  Of 
three  quantities  in  II.  P.,  the  middle  one  is  called  the  harmonical 
mean  of  the  other  two. 

If  a,  b,  c  are  in  II.  P.,  then  -»  t.  -  are  in  A.  P.,  whence 

ci    0    c 

d       a~  c       h  '     1)  '~  a       c* 
Hence  the  harmonical  mean  b  ol  a  and  c  is 

2ac 


I 


a  -\-  0 


To  insert  71  harmonical  means  between  a  and  c,  we  construct 
t'.ie  11.  P.  of  71  -f-  2  terms  of  which  a  is  the  first  and  c  the  last 
term.  For  example,  to  insert  4  harmonical  means  between  1  an.d 
•I-,  we  observe  that  1,  2,  3,  4,  5,  6  are  in  A.  P.  and  therefore  that 
1^  h  h  h  Ty  i  are  in  H.  P. 


72  HARMONICAL   PROGRESSION.  [Cii.  VI 

EXERCISES. 

1.  The  geometrical  mean  of  any  two  quantities  is  also  the  geometrical 
mean  between  their  arithmetical  and  harmonical  means. 

2.  The  conditions  that  a,  b,  c  may  be  in  A.  P.,  G.  P.,  or  H.  P.  are  i 

a  —  h  '.!)  —  c  '.'.  a  :  a, 
a  —  b  :  b  —  c  ::  a  :  b, 
a  —  b  :  b  —  c  ::  a  :  c,  respectively. 

3.  If  a,  b,  c  are  in  G.  P.,  then  a  -{-  b,  2b,  b  -{-  c  are  in  H.  P. 

4.  If  a,  b,  c  are  in  H.  P.,  then  a,  a  —  c,  a  —  b  are  in  H.  P. 

6.  If  x^,  y\  2'  are  in  A.  P.,  then  y  +  2,  z  -\-  x,  x  -{-  y  are  in  H.  P. 

6.  If  a,  b,  c  are  in  H.  P.,  then  2a  —  b,  b,  2c  —  b  are  in  G   P. 

7.  If  a,  b,  c  are  in  A.  P.  and  b,  c,  d  are  in  H.  P.,  then  a  :  b  =  c  :  d 

8.  If  three  distinct  numbers  a,  b,  c  are  in  A.  P.,  while  a\  b^,  c^  are  in  H  P.^ 

then  —  —,  6,  c  are  in  G.  P. 

9.  Insert  four  harmonical  means  between  |  and  f^  ;  between  2  and  Y 


CHAPTER  VII. 
COMPOUND   INTEREST   AND   ANNUITIES. 

71.  Problem :  To  find  the  amount  A  of  a  given  sum  or  principal 
P  at  compound  interest  for  n  years  at  the  rate  of  R  per  cent  a  year. 

To  simplify  the  formulae,  set  r  =  R/100.  The  amount  of 
both  principal  and  interest  at  the  end  of  the  first  year  will  be 
P  +  Pr  —  P{1  +  r).  At  the  end  of  the  second  year,  the  amount 
will  be  {P(l  +  r) }  (1  +  r)  =  P(l  +  r)\  At  the  end  of  the  third 
year,  the  amount  will  be  P(l  +  r)^,  etc.  Hence  at  the  end  of  n 
years,  the  amount  will  be 

A  =  P{1  +  r)\ 

If  the  interest  be  compounded  semi-annually,  the  amount  at  the 
end  of  n  years  is  found,  by  a  similar  argument,  to  be 

If  the  interest  be  compounded  quarterly,  tKe  amount  will  be 

Example.  Find  the  amount  of  $1000  in  20  years  at  compound  interest  at 
6  per  cent  a  year,  payable  semi-annually. 

Since  P  =  1000,  n  =  20,  r  =  .06,  the  amount  is  A^  =  1000(1. 03)«. 
By  the  four-place  logarithmic  table,  pp.  24,  25,  we  have 

log  1.03  =  0.0128. 

.-.  log  A^  =  log  1000  +  40  log  1.03  =  3.5120. 

.-.^2  =  3251, 

73 


74  COMPOUND  INTEREST  AND  ANNUITIES.  [Ch.  YII 

the  last  figure  being  wholly  unreliable.     With  a  seven-place  table, 

log  1.03  =  0.0128372, 
and  log  A^  =  3.5134880,    whence  A^  =  3262.03. 

72.  Problem:  To  find  the  present  value  P  of  a  sum  A  which  is 
to  be  paid  at  the  end  of  n  years,  allowing  compound  interest  at  R 
per  cent.     Set  r  =  R/IOO. 

The  amount  of  P  at  the  end  of  n  years  must  equal  A.     Hence 
P-A{1  +  r)-^ 

73.  A  fixed  sum  of  money  paid  annually  is  called  an  annuity. 
The  subject  finds  immediate  application  in  Insurance.  The  sim- 
plest problem  in  annuities  is  the  following  : 

To  find  the  amount  A  of  an  annuity  a  allowed  to  accumulate 
for  n  years  at  compound  interest. 

The  first  annual  installment  a  is  due  at  the  end  of  the  first  year 
and   hence   draws  interest  for  n  —  1  years;    its  amount  will   be 
a(l  +  ^)"~^.     The  second  annual   installment  a  will  amount  to 
a{l  +  rY~^,  etc.     The  last  installment  a  draws  no  interest.    Hence, 
the  amount  of  the  n  installments  will  be 

^  =  •«  +  «  (1  +  r)  +  «(1  +  rf  +  .  .  .  -f  a(l  +  r)"-2+  a{l  +  rY'^ 

by  the  formula  for  the  sum  of  geometrical  progression  of  ratio 
1  +  r. 

74.  Problem :  To  find  the  present  value  of  an  annuity  a  to 
continue  n  years,  allowing  compound  interest. 

By  §  72,  the  present  value  of  a  due  one  year  hence  is  a{l-\-r)~'^] 
the  present  value  of  a  due  two  years  hence  is  ^^(1  +  ^*)  ~^?  •  •  •  j  ^^^^ 
present  value  of  a  due  n  years  hence  is  a(l  +  r)T"".  The  total 
present  value  is 

a{l  +  r)-'-{-a{\+r)-^-\-  .  .  .  +«(l  +  r)-« 


Sec.  74]  COLLEGE  ALGEBRA.  75 

Since  the  ratio    (1  +  r)~^  is  less  than  unity,   the   geometrical 

progression    may  be    summed    to    infinity,   giving  - .      Hence  the 

present  value  of  a  perpetuity  (perpetual  annuity)  a  is  —,  the  rate 


of  comDOund  interest  as  above.    Thus  a  perpetuity  of  $100  has  the 
100 
.04 


presen+^  value  — r^  =  2500  dollars,  if  compound  interest  is  reckoned 


at  4  per  cent. 

EXERCISES. 

[The  calculation  of  (1  +  ^)"  for  r  small  and  n  large  by  means  of  our  four- 
place  table  of  logarithms  is  not  very  accurate.     By  a  six-place  table, 

log  1.02  =  .008600,         log  1.03  =  .012837,         log  1.04    =  .017033, 
log  1.05  =  .021189,         log  1.06  =  .025306,         log  1.025  =  .010724.] 

1.  Find  the  amount  of  $1000  in  30  years  at  6  per  cent  compound  interest, 
the  interest  being  paid  annually. 

2.  What  sum  of  money  will  amount  to  $900  in  1^  years  at  4  per  cent  com- 
pound interest,  interest  being  compounded  semi-annually  ? 

3.  What  is  the  present  worth  of  $5000  due  in  10  years,  allowing  compound 
interest  at  5  per  cent  paid  semi-annually  ? 

4.  In  how  many  years  will  a  sum  of  money  double  itself  at  5  per  cent 
compound  interest,  due  annually  ? 

5.  Find  the  amount  of  an  annuity  of  $1000  in  20  years,  allowing  com- 
pound interest  at  3  per  cent  per  annum. 

6.  What  is  the  present  value  of  an  annuity  of  $500  for  30  years,  allowing 
interest  at  5  per  cent  compounded  semi-annually  ? 

7.  A  person  borrows  $10000  at  5  per  cent  compound  interest.  How  much 
must  he  pay  in  annual  installments  in  order  that  the  whole  debt  may  be  paid 
in  15  years? 

8.  What  should  I  pay  for  a  perpetuity  of  $500  to  begin  10  years  hence,  if 
compound  interest  is  reckoned  at  3  per  cent  ? 


CHAPTER  VIII. 
UNDETERMINED   COEFFICIENTS;    PARTIAL  FRACTIONS. 

76.  A  variable  quantity  is  one  which  may  assume  different 
values  (usually  an  unlimited  number  of  values)  in  the  same  investi- 
gation. A  constant  quantity  is  one  which  retains  the  same  value 
throughout  the  discussion. 

Thus  in  a  given  geometrical  progression  whose  first  term  is  a 
and  common  ratio  is  r,  the  Tith  term  4  is  rtr"~^  We  consider  a  and 
r  to  be  constants,  while  n  is  a  variable  ;  by  giving  to  n  the  values 
1,  2,  3  4,  .  .  .  ,  in  succession,  we  obtain  for  t^  the  values  a,  ar,  ar^, 
ar^,  .  .  .  ,  respectively. 

The  area  of  a  circle  of  radius  E  is  ttEK  To  obtain  different 
circles  we  let  E  vary;  but  tt  is  a  constant  number. 

A  variable  y  is  called  a  function  of  a  variable  x  if  to  every  value 
that  may  be  assigned  to  x  there  corresponds  a  definite  value  of  i/. 
It  sometimes  happens  that  certain  isolated  values  may  not  be  as- 
signed to  X,  since  y  then  ceases  to  have  a  definite  value  (see  §  100). 

In  the  preceding  examples,  t^  =  ar"~Ms  a  function  of  7i,  the 
area  ttE^  of  the  circle  is  a  function  of  the  radius  E.  Similarly 
a:*,  3x  -\-  1,  ^x^  log  x  are  functions  of  x. 

It  is  often  convenient  to  employ  symbols  which  emphasize  the 
functional  dependence.  Thus,  for  asmd  r  constant  and  7i  variable, 
we  used  /"*  to  denote  the  function  «r"  ~  ^  of  w.  Likewise,  ^x  and 
log  X  are  functional  symbols. 

76 


Sec.  76]  COLLEGE  ALGEBRA.  77 

76.  By  formulae  (8)  and  (9)  of  §  31,  we  have  the  identity 

(1)  ax^  -{-  hx  -\-  c~  a{x  —  a)(x  —  /?), 

if  a  and  /3  denote  the  roots  of  the  equation  ax'^  -\-  bx  -\-  c  =  0. 
To  generalize  this  result,  consider  the  expression 

U^  ~  ax""  +  bx""-'  +  cx""-^  +  .  .  .  +kx  +  l, 

called  a  rational  integral  function  of  x  of  degree  7i.  If  the  ex- 
pression E^  vanishes  when  a  is  substituted  for  x,  then,  by  the 
factor  theorem,  x  —  a  is  'd  factor  of  B^,  The  first  term  of  the 
quotient  Q  obtained  upon  dividing  B^hj  x  —  a  is  clearly  ax^~^. 
Hence 

F,=  {x-a)Q,     Q  =  ax^-'+... 

Similarly,  if  Q  vanishes  when  ^  is  written  for  x,  then 

Q  =  {x-/3)Q\     Q'^ax^-'+,.. 

Hence  F^=  {x-  a){x  -  /3)Q\ 

so  that  E^  vanishes  when  ^  is  substituted  for  x.  7/"*  there  be  n 
distinct  valued  a^  ^,  ,  ,  , ,  v  of  x  for  which  E^  vanishes,  a  continua- 
tion of  the  preceding  argument  shows  that  E^  has  the  7i  factors 
X  —  a,  x  —  /3,  .  ,  ,  y  X  —  V  and  a  final  numerical  factor  a,  so  that 

(2)  E,  =  a{x  -  a){x  -  (3)  .  .  .  {x  -  v). 

Then  a,  /3,  .  ,  ,  ,  v  are  roots  of  the  equation  E^  =  0. 

77.  Theorem.  A  ratmial  integral  fu7icti07i  of  the  nth  degree  in 
X  cannot  vanish  for  more  than  n  values  of  x^  unless  the  coefficients 
of  all  the  poivers  of  x  are  zero. 

Suppose  that  the  function  E^  vanishes  ior  n  -\-l  distinct  values 
X  and  a,  ft  ,  .  . ,  V  oi  x.  Then  E^  must  have  the  form  (2),  and 
must  moreover  vanish  for  x  =  \.     Hence 

a{X  -  a){X  ^  /3)  .  ,  .  {X  -  r)  =  0. 

*  The  question  of  the  existence  of  such  values  a,  /jf,  .  .  .  ,  is  considered  in 
Chapter  XIX. 


78   UNDETERMINED  COEFFICIENTS;  PARTIAL  FRACTIONS.  [Cii.  VIII 

Since  the  differences  X  —  a,  .  .  ,  ,  X  —  r  all  differ  from  zero,  a 
must  be  zero.  Hence  B^  reduces  to  hx""  ~'^  -\-  cx^"^-]-  .  .  .  Since 
this  function  of  degree  7i  —  1  vanishes  for  more  than  n  —  1  values, 
b  must  be  zero.  Similarly,  c  =  Oy  .  ,  .  ,  k  —  0,  I  —  0. 
Another  statement  of  the  theor-em  is  the  following: 
A7i  equation  of  degree  n  cannot  have  more  than  7i  roots  unless 
all  the  coefficients  are  zero. 

78.  Suppose  that  the  two  functions  of  degree  n 

are  equal  in  value  for  more  than  n  values  of  x.     Then 

vanishes  for  more  than  n  values  of  x.     By  the  previous  theorem, 
the  coefficients  of  all  the  powers  of  x  must  be  zero,  whence 

Po  =   ^0  '       ]\^   9l^        •    •    •  ^        Pn-l  =  qn-l,       Vn  =  9^ 

If  ttuo  rational  i)degral  functions  of  the  nth  degree  in  x  he  eqnal 
in  value  for  more  than  n  values  of  x,  the  coefficients  of  like  potuers 
of  x  are  equal. 

In  particular,  if  two  rational  integral  functions*  of  -x  each  of 
iinite  degree  are  equal  in  value  for  all  vahies  of  x,  the  coefficients 
of  like  powers  of  x  may  be  equated  to  each  other. 

79.  To  illustrate  the  application  of  the  preceding  theorems  to 
problems  on  integral  functions,  we  consider  certain  examples. 

Example  1.  Find  the  conditions  on  a,  h,  c,  d  in  order  that  a^y^  +  ^^^ 
-\-  ex  -\-  d  may  be  a  perfect  cube  for  all  values  of  x. 

It  must  be  the  cube  of  an  expression  Ix  -\-  m  oi  the  first  degree,  in  which 
I  and  m  are,  as  yet,  undetermined  coefficients.     Set  * 

aa?  -\-  hx^  -\-  ex  ■{-  d~{lx  -\-  mf  ~  Px^  +  Sl'hnx'^  +  dlm^x  -\-  m\ 
Since  the  first  and  third  expressions  are  equal  for  all  vahies  of  x,  ' 

a  =  l^,     b  —  SPm,     c  =  Slm\     d  =  m^. 
Hence  I  —  a^,  m  —  d^.     The  second  and  third  conditions  then  give 
h  =  ^a^dK     c  =  3a^c?l 

*  The  symbol  E  is  employed  in  identities.  Here  the  relation  is  an  identity 
in  x. 


Sec.  79]  COLLEGE  ALGEBRA.  79 

Inversely,  if  h  and  c  have  these  values,  then 

ax^  -[-  bx^  -\-  ex  -\-  d  =  {a^x  -j-  d  ^f. 
Example  2.     Determine  the  value  of  k  so  that  x^  4-  ^x^  ^  ^x  -]-  k  shall 
be  divisible  hy  x^  -{■  x  -{-  1. 

The  quotient  must  have  the  form  x"^  -}-  7ix  -f-  k.     Set 

x^  +  ^x^  f  3.C  +  ^  =  («2  j^  x-\-  l){x^  -\-hx^  k). 
Expanding  the  right  member  and  equating  coefficients  of  like  powers  of  x 
in  the  two  expressions,  we  have  the  conditions 

0  -r  7i  -}-  1,     4  =  ^^  -f  /i  -I-  1,     'd  --^h^  k, 

which  are  all  satisfied  if  ^  —  —  1,  ^  =  4. 

»  Example  3.     Find  the  sum  of  the  squares  of  the  first  n  integers. 
Since  the  sum  depends  upon  n,  we  assume  that 
-I-  22  -{-  .  .  .  +  (7^  -  1)2  +  ?i2  =  ^  +  hn  -f  en'  +  dii^  +  en^  -{- fn^  +  .  .  ., 
wnere  a,  b,  c,  d,  e,  f,  .  .  .  are  undetermined  coefficients,  each  independent  of  n. 
Changing  n  into  n  -\-  i,  we  get 

124-224- .  .  .-^n'-j-{ni-lf  =  a+b{n+l)-i-c{n-\-iy-]-d{n-^lf^e{ni-iy+.  .  . 
Subtracting  the  previous  identity,  we  get 

n^-\-2n^l~b  +  2cn  +  c-\-  Mn^  +  Mn  -^  d  ^  Aen^  +  6en^  -{- ien  +  e -i-  .  .  , 
Since  this  equation  holds  for  every  value  of  n,  the  coefficients  of  like  powers 
of  n  may  be  equated.  Hence  e  =  0  and  /,  and  all  succeeding  coefficients, 
must  be  zero.     Also, 

dd  ^  1,     3^  +  2c  =  2,     (Z  +  c  +  5  =  1, 
wheuce  d  =  I,  c  —  I,  b  =  I.     Hence 

12   +    22   +    .    .    .    +    (?l   -    1)2   -f    ?,2  =  ^  4_    1^^   4_   J^2   _|_    J^3^ 

If  71  —  1,  there  is  a  single  term  1  in  the  series,  so  that 
1  =  ^  +  J  +  i  +  *,     or     a  =  0. 

Hence  I2  +  22  +  .  .  .  +  712  z=  ^n{n  +  l){2n  +  1). 

EXERCISES. 

1.  When  is  a'^x^  +  bx^  -\-  ex'  -\-  dx  ~\- p  a  perfect  square? 

2.  Find  the  condition  that  x^  —  Spx  -f-  2q  may  have  a  factor  of  the  form 
{x  -  c)2. 

3.  Find  k  so  that  there  are  solutions  x,  y,  z  not  all  zero  of 
X  -{-  y  —  z  =  0,     2x  —  y  -{-  dz  —  0,     x  -\-  ky  -\-  z  =  0. 

4.  If  ax^  +  bx^  -}-  ex  -\-  d  is  divisible  by  x''  +  h,  then  rfd  —  be. 
Using  undetermined  coefficients,  establish  the  sums 

5.  1  +  2-^3  +  ...+^==^  i^'in  +  1). 

6.  12  4-  32  4-  52  +  ...  -f  {2n  -  If'  =  ln{^v?  -  1). 


8o  UNDETERMINED  COEFFICIENTS;  PARTIAL  FRACTIONS,  [Ch.  VII 

7.  1^  +  2^  +  33  +  .  .  .  -f-  71^  =  ln\n  +  1)2. 

8.  13  +  33  +  53  +  .  .  .  +  (271  -  \f  =  n\2n''  -  1). 

9.  1*  -h  2*  +  3*  +  .  .  .  +  71*  =  ^\n{n  +  l)(27i  +  l)(37i2  -^  37i  -  1). 

10.  1-2  +  2-3  +  3.4  +  .  .  .  +  n{n  +  1)  =  M^i  +  1)(^  +  2). 

11.  1.2-3  4-  2.3.4  +  3.4.5  +  .  .  .  +  n(n  +  l)(n  +2)  r=:  ^n(7i+l)(7i+2)(;2+3)r 

12.  The  sum  of  the  cubes  of  the  first  n  integers  equals  the  square  of  their 
sum. 


PARTIAL   FEACTIONS. 


I 


80.  It  is  desirable  to  be  able  to  express  a  complex  fraction  as  the 
sum  of  simpler  fractions,  called  partial  fractions.     Thus 

/3^  4-5:.        _      1  « 

\*^I  1    _   Q^     I      0V2    ~~   1     „   ^  "I 


1  —  3a;  +  2a;2       1  —  a;  '   1  ~  2a;* 

To  expand  the  complex  fraction  in  the  left  member  into  a  series  of 
ascending  powers  of  x,  we  may  expand  the  partial  fractions  on  the 
right  and  add  the  resulting  series.  Indeed,  by  the  theory  of  geo- 
metrical progressions,  if  r  be  numerically  less  than  unity,  the  fol- 
lowing expansion  holds: 

=  1  +  r  +  r2  +  r^  +  . .  . 


1  -  r 
Hence,  if  2x  is  numerically  less  than  unity, 

i  —  X 

3(1  +  2a;  +  4a;2  +  .  .  .  +  2*^a;"  +  ...), 


l-2a; 
4. /it* 

81.  To  explain  the  general  method  employed  to  decompose  into 
partial  fractions  a  given  complex  fraction  whose  denominator  is  of 
higher  degree  in  x  than  the  numerator,  consider  the  fraction  in  the 

left  member  of  (3).      We  observe  that  the  fractions and 

^  ^  1  —  X 


I 


Sec.  82]  COLLEGE  ALGEBRA.  8l 

r — ,  when  reduced  to  the  denominator   (1  —  x){l  —  2x)^  will 

1  —  tlx 

contribute  to  the  formation  of  a  complex  fraction  with  the  given 

denominator  and  with  a  numerator  of  the  first  degree,  while  no  new 

fraction  witli  a  denominator  of  the  first  degree  will  have  this  prop- 

i  erty.     Hence  if  our  aim  is  to  be  attained,  the  decomposition  must 

be  of  the  form 

4:  —  5x  ^1  ^ 


1  -'dx  +  'Zx^       1  -  a;  ^  1  -  2a;' 

in  which  the  undetermined  coefficients  a  and  b  are  to  be  determined. 
If  this  be  possible,  we  shall  have 

4t-bx  =  a{l  -  2x)  +  b{i  -x)^a  +  h  -  x{2a  +  b). 

In  view  of  its  origin,  this  relation  is  to  be  true  for  all  values  of  x 
except  1  and  ^.  Then,  by  §  78,  the  relation  must  be  true  for  all 
values  of  x,  and  the  constant  terms  as  well  as  the  coefficients  of  x 
may  be  equated.     Hence 

rt -f  ^  =  4,     2a  +  b  =  5, 

Hence  «  =  1,  ^  ~  3,  giving  the  true  result  (3). 

82.  As  a  second  example,  we  decompose  into  partial  fractions 

4:X^  +3X-1 

{X-  iy(x  +  2y 

It  is  clear  that  fractions  with  the  denominators  a;  -\-2,x  —  l,(x  —  1)^ 
and  with  constants  for  numerators  may  contribute  to  the  formation 
of  a  fraction  with  the  denominator  (x  +  2){x  —  1)^  and  with  a 
luinierator  of  degree  at  most  two.  There  is  nothing  gained  by 
employing  also  fractions  of  the  forms 

mx  -\-  n  rx  -\-  s 

{x^2){^^'      (^^^Tp' 

since  they  are  themselves  expressible  as  sums  of  partial  fractions, 

\{rn  +  n)       \{2m  -  n)  r  s -^  r 

x-l     "^      x-\-2     '     x-l'^{x-iy^' 


82    UNDETERMINED  COEFFICIENTS ;  PARTIAL  FRACTIONS,  [Ch.  YIHI 

respectively.     Moreover,  -, — — r-^  cannot  be  expressed  as  the  sum* 

of  fractions  with  denominators  of  the  first  degree.  The  decomposi- 
tion will  therefore  give  partial  fractions  of  simpler  form  than  the 
given  fraction  only  when 


4a;2  -)-  3^  -  1 


+  ■ 


(x  -  1)2(^+2)       x  +  2^  x-l^  {x  -  If 
for  suitable  values  of  the  undetermined  coefficients  a,  t,  c.     Then 
(4)     4a;2  +  3a;  -  1  =  a{x  -  If  +  h(x  -  l){x  +  2)  +  c[x  +  2). 
Equating  the  coefficients  of  x^,  of  x,  and  the  constant  terms,  we  get 
a  +  Z>  =  4,      -  2a  +  /^  +  c  =  3,     a  —  2h+2c=  -  1. 


I 


Solving  by  determinants  or  otherwise,  we  get  «  =  1,  Z>  =  3,  c  =  2, 
A  simpler  method  of  determining  a,  Z>,  c  is  to  substitute  special 

values  for  x  in  (4).     We  observe  that,  in  view  of  §  78,  relation  (4) 

is  an  identity  and  is  there  fore,  true  for  the  values  x=\,  —  2,  etc. 

For  X  —  \,    we  get   G  =  3c  ;    for  a;  =  —  2,    9  =  Ort  ;    for  x  —  0, 

—  l  =  a  —  U  +  2c.     Hence  6?  =  2,  a  =  l,  b  =  3. 

83.  The  presence  of  imaginary  numbers  offers  no  difficulty.     To 

decompose  the  following  fraction,  we  set 

42  —  19a;  ^      J-  ^  •  ^ 


{x-4:){x^  +  l)       x-4:       x^\/'-\       x-\/'-\ 
.-.     42-19.^  =  a{x^-^\)-^l{x-^){x-\/~^)-^c{x-^)(pc^\^~^). 

Equating  coefficients,  we  obtain  the  values  of  a,  h,  c  given  below. 
With  the  explanation  made  at  the  end  of  §  82,  we  may  simplify  the 
work.     For  a;  =  4,  we  get  —  34  =  11a,   whence  «  =  —  2.     For 


x 


-/  —  1,  we  get 


42  -  194/  -  1  n=  c(4/  -  1  -  4)  (2|/  -  1),     c=l-\-  VV  -  1. 


Sec.  84]  COLLEGE  ALGEBRA.  83 


For  X  =  —  \/  —  I,  we  get   h  —  1  —  ^^-\/  —  1.     Hence 


42-192:  -2,1-  -VV  -  1    ,    1  +  VV  -  1 


(^  _  4)(:^^  +  1)       x-4.   '      x  +  \/  -l  x-\/  -I 

The  second  and  third  partial  fractions  are  conjngates,  since  one 
is  derived  from  the  other  by  replacing  4/  —  1  by  —  |/  —  1.     Hence 

2x  —  11 
their  sum  must  be  real  (§  4).     In  fact,  their  sum  is    \  . 

We  might  have  avoided  the  introduction  of  imaginaries  by  setting 

42  -  19.T  a       ,   ex-^f 


42  -  192:  =  a{x^  +  1)  +  {ex+f){x  -  4). 

Observing  that  this  relation  is  an  identity  by  §  78,  we  may  set 
4,  whence  a  =  —  2.  Equating  the  coefficients  of  x^  and  of  x, 
we  get 

6  =  2,     -  19  ==  -  4^  +/,    whence /=  -  11. 

84.  If  the  degree  of  the  numerator  of  the  given  fraction  equals 
ar  exceeds  the  degree  of  the  denominator,  w^e  first  divide  the  nu- 
merator by  the  denominator  and  obtain  a  remainder  of  degree  less 
than  the  degree  of  the  denominator.     For  example, 

122:^  +  102:2  _  14  iQ:c  -  8 

=  42:  +  6-' 


3^C2  _  9;^  _   1  '  '      3^2  _  22:  —  1  ' 

Proceeding  as  usual,  the  second  fraction  may  be  decomposed  into 

IO2:  2 


'dx  -\-  1        2:  —  1 

The  division  mentioned  may  be  avoided  by  assuming  that 

122:3  +  102:'^- 14         ,      ,         ,         ^^^         ,        ^ 
o2r  -  2x  —  1  '         '    32:  +  1       2:  —  1 

ind  determining  «,  h,  c  by  the  usual  method. 


84  UNDETERMINED  COEFFICIENTS;  PARTIAL  FRACTIONS,  [Ch.VIII 

EXERCISES. 
Resolve  into  partial  fractions: 


7aj-  1 
a?  -V 

ar'  -f  15aj  +  8 

(X- 

x"  - 

-  ^){x  +  1) 

-  3aj  4-  6 

aj*+  aj2-f-  2 

*  a;(aj  -  l)^ 
°-   (05  -f  b){x'  +  1)* 

10    -     ■    -     '    -  11    ^^'  -  ^  +  1  12 

aj*  +  1      '  •       (a;  -  If    •  ■'^-  (ar*  -  l){x  +  If 

13.  Expand  into  series  the  fractions  of  Exs.  1,  3,  5,  7,  9. 


1  -  5a;  4-  6a;2* 

^x"  -x-{-% 

.  a^4-«     • 

2a^-3 

x'-^l' 

5aj3  4-  6a;2  +  hx 

i 


CHAPTER  IX. 


PERMUTATIONS  AND   COMBINATIONS;    BINOMIAL   AND   MULTI- 
NOMIAL THEOREMS  FOR  POSITIVE  INTEGRAL  INDEX. 

85.  Three  letters  a,  l^  c  may  be  arranged  in  six  ways 

ahc^     acl,     lac^     tea,     cab^     cba, 

Jhile  there  are  only  three  groups  (or  selections)  each  of  two  letters 
isen  from  three  letters  a,   Z>,  Cy  viz.,  ab^  ac,  be,  we  obtain  six 
arrangements  of  three  letters  two  at  a  time,  namely, 

ab,     ba,     ac,     ca,     be,     cb. 

Definitions.  Each  of  the  arrangements  which  can  be  made 
with  r  things  chosen  from  n  things  is  called  a  permutation  of  the  n 
things  r  at  a  time.  Each  of  the  groups  (selections)  which  can  be 
made  by  selecting  r  things  from  7i  things  is  called  a  combination 
of  the  n  things  r  at  a  time. 

Thus  there  are  six  permutations  of  a,  b,  c  two  at  a  time : 

ab,     ba,     ac,     ca,     be,     cb. 

There  are  three  combinations  of  a,  b,  c  two  at  a  time :  ab,  ae,  be. 

In  a  combination,  the  order  in  which  the  letters  are  written  is 
indifferent;  in  a  permutation,  the  order  is  essential.  Thus  the 
pairs  ab  and  ba  give  the  same  combination,  but  give  distinct  per- 
mutations. 

-     _  P^ 


S6  PERMUTATIONS  AND   COMBINATIONS,  [Cn.  IX 

86.  The  7mml)er  of  jjermtitalioiis  of  n  different  tilings  r  at  a 
time  is  n{n  —  l){n  —  2)  .  .  .  (/^  —  r  +  1).* 

AVe  are  to  find  the  number  of  ways  in  which  we  can  iill  r  places 
when  we  have  n  different  tilings  at  our  disposal. 

For  the  first  place  we  may  take  any  one  of  the  n  things;  for  the 
second  place  any  one  of  the  remaining  n  —  1  things;  for  the  third 
place  any  one  of  the  now  remaining  n  —  2  things,  etc.  Hence  the 
r  places  may  be  filled  in  n{7i  —  l){?i  —  2)  .  .  .  {71  —  r  -j-  1)  ways, 
since  the  rth  factor  is  71  —  (r  —  1). 

Corollary.  The  number  of  permutations  of  71  different  things 
taken  all  at  a  time  is  71(71  —  l)(?z  —  2)  .  .  .  3 •2-1.  This  product 
of  all  the  natural  numbers  up  to  and  including  71  will  be  denoted 
by  the  symbol  7i\  which  is  read  ''71  factorial."  f 

For  example,  there  are  3!  =  G  permutations  of  a,  I),  c  three  all 
a  time  (given  in  §85)  and  3-2  =  0  permutations  of  a^  Z>,  c  two  alj 
a  time  (given  in  §  85). 

87.  We  notice  that  each  of  the  three  combinations  al),  ac,  be  o\ 
the  letters  a^  b,  c  taken  two  at  a  time  furnishes  exactly  two  permid 
tations  of  a,  b,  c  two  at  a  time;  thus  the  combination V/Z>  furnishes 
the  two  permutations  ab  and  ba.  Moreover,  the  six  resulting  per- 
mutations give  alt  the  permutations  of  «,  b,  c  two  at  a  time.  Simi- 
lar remarks  hold  true  in  the  general  case  next  considered. 

Let  „6r  denote  the  number  of  combinations  of  71  different  thing} 
r  at  a  time.  Let  ^P^  denote  the  number  of  permutations  of  7 
different  things  r  at  a  time.     AVe  have  shown  that 

^F^=n{n-l){n-2).  .  .  (m  -  r  +  1). 

Each  one  of  the  ^C^  combinations  consists  of  a  set  of  r  differen 
things,  which  may  therefore  be  arranged  or  permuted  in  exactly 
rPr  =  t\  distinct  ways.  Every  such  arrangement  is  a  permutatioi 
of  the  71  things  taken  r  at  a  time.     By  starting  with  a  suitabh 

*  The  number  is  zero  if  r  >  n.     The  product  then  has  a  factor  71  —  n  =  0 
f  The  symbol  I  n  is  also  used. 


Sec.  88]  COLLEGE  ALGEBRA.  87 

combiiuitioii  of  r  of  the  things  and  arranging-  tliem  in  a  suitable 
way,  we  may  rer.ch  any  given  pei'mutation  of  the  n  things  r  at  a 
time.     Hence 

„(7,  X  r\  =  „P,  =  n{n  -  1)  ,  ,  ,  (n  -  r  +  1). 
.       r  -  n{n-l){n-2).  ^jjn^^  +  1) 
•   •  "^  1-2-3  .  .  .  r 

Upon  multiplying  the  numerator  and  the  denominator  by 
{7i  —  r){n  —  r  —  1)  .  .  .  3  •  2  •  1  =  (m  —  r) ! ,  we  get  * 

C  -  '''' 

88.  The  nnmher  of  comMnatioiis  of  n  different  things  r  at  a  time 
eqiials  the  nnmher  of  comhinations  of  n  tilings  n  —  r  at  a  time. 

When  a  set  of  r  things  is  selected  from  01  things,  there  is  left  a 
set  of  n  —  r  things.  Moreover,  any  given  set  of  n  —  r  things  may 
be  left  (negative  selection)  by  making  a  suitable  selection  of  r 
th ings.     Hen ce  „  C,.  ~  ^  C,,  _ ,. . 

The  proposition  also  follows  from  the  formula  for  ,,(7^..     Thus 


ri '_:_: :  _         "•         _    p 

"  -^'  -  (;,  _  r) !  [;/  -  {n  -  r)]  !       {n  -  r) !  r\  ~  ""   '" 


n^n 


89.  Theorem.      For  positive  integers  n  mid- r, 

The  combinations  of  n  +  1  different  letters  a^^  a^,  •  •  •  ?  «w> 
^^i  +  i  ^'  ^t  a  time  may  be  separated  into  two  sets,  according  as  the 
combination  contains  the  last  letter  <7,,  +  i  or  does  not.  If  the  last 
letter  be  taken,  there  remain  only  r  —  1  letters  to  be  selected  from 

*For  r  =  71,  the  fir.it  result  gives  nCn  —  —  =  1,  w^liile   the   second  gives 

n\ 
,iCa  —  — ~.     The  second  formula  therefore  holds  for  r  =  n  only  when  we 

adopt  the  notation  0!  =  1.     Similarly,  the  result  in  §88  holds  for  r  =  n  only 
when  we  put  uCq  ~  1. 


0^0 

>c„ 

A. 

A 

A. 

A. 

A 

A. 

,0, 

A, 

Ao 

fi. 

A 

A, 

88  PERMUTATIONS  AND   COMBINATIONS.  [Ch.  IX 

the  n  letters  «j,  a^,  ,  ,  .,  a^^  which  can  be  done  in  nCr-i  ways. 
If  the  last  letter  be  not  taken,  we  must  select  r  letters  from  the  n 
letters  «j ,  rt^,  .  .  .,  a„ ,  which  can  be  done  in  ^0^  ways.  The  sum 
of  the  two  numbers  n^r-i  ^^^  n^r  must  equal  n  +  i^^r- 

The  formula  remains  true  for  r  =  1,  if  we  take  ^G^  =  1,  there 
being  one  way  of  selecting  no  objects  from  n  objects. 

An  interesting  application  is  the  construction  of  Pascal's 
Triangle: 

11  --^ 

1     2^  1 
1     3     3     1 
,C,  14     6     4     1 

The  above  formula  now  shows  that  any  number  n  +  i^^r  equals  the 
sum  of  the  number  ^0^  just  above  it  ancj  the  number  „(7,._i  to  the 
left  of  „(7^.  For  example,  ^C^  =  ^C,  +  ,C\,  or  6  ==  3  +  3.  The 
numbers  in  the  next  row  of  the  table  on  the  right  would  there- 
fore be 

1     5     10     10     5     1. 
It  appears  that  the  numbers  in  the  oith  row  of  the  table  are 
the  binomial  coefficients  in  the  expansion  of  (1  +  ^Y  [§  91]. 

90.  We  proceed  to  determine  the  number  of  permutations  of  n 
things  taken  all  at  a  time,  when  the  things  are  not  all  different. 
There  are  only  three  permutations  of  a,  a,  h  taken  three  at  a  time, 
namely,  aah,  aba,  laa.  If  we  replace  the  two  a's  by  two  distinct 
letters  a^  and  a^ ,  the  first  permutation  aal)  will  furnish  two  permu- 
tations a^ajb  and  a^afi.  In  this  way,  we  reach  the  3-2  =  6  permu- 
tations ot  a^y  a^,  h. 

In  general,  let  there  be  n  letters,  p  of  which  are  a^s,  q  of  which 

are  <5>'s,  and  r  of  which  are  c's,  so  that  n  =p  -^  q  -\-r.      Let  P 

denote  the  required  number  of  permutations  of  these  n  letters  n  at 

a  time.     Consider  any  one  of  the  permutations  as 

aa  .  ,  ,  abb  .  .  .  bcc  .  .  .  c. 


Sec.  90]  COLLEGE  ALGEBRA.  89 

If  we  replace  the  p  letters  a  by  2^  distinct  letters  «i ,  ^2 ,  .  .  .  ,  a^, 
we  may  derive  from  the  given  permutation  jr?!  permutations 

a^a^ .  .  .  apb  ,  .  .he ,  .  ,  Cy     a^a^ .  .  .  aj) .  .  ,  he ,  ,  .  c^  etc. 

Hence  from  the  P  permutations  of  the  «'s,  ^'s,  c's,  we  derive  P  ,p\ 
permutations  oi  a^,  a^.,  .  .  .  ^  a^,  h,  .  ,  ,  ,  h,  c^  .  .  , ,  c.  Similarly,  if 
in  one  of  the  new  permutations  we  replace  the  q  letters  bhj  q  dis- 
tinct letters  h^^  h^,  .  .  .  ^  h^,  we  may  obtain  exactly  q\  permutations 

a^a^ .  .  .  aphj)^ ,  .  .h^c .  .  .  c,     a^a^ .  .  .  a^h^h^ .  .  .h^c  .  .  .c,  etc. 

Hence  the  P .  p\  permutations  of  a^,  a^, , .  . ,  ap,  h,  .  ,  ,  ^h^  c^ ,  ,  ,  ,c 
give  rise  to  exactly  P , p\  q\  permutations  of  a^,  a^,  ,  , , ,  ap,  J^, . .  .  , 
hq,  c,  .  .  .  ,  c.  Finally,  if  the  r  letters  c  be  replaced  by  r  distinct 
letters  Cj,  c^,  .  .  . ,  c^,  the  resulting  number  of  permutations  on  the 
p  -\-  q  -\-  r^n  distinct  letters  will  be  P,p\q\r\  But  the  num- 
ber of  permutations  of  n  distinct  letters  n  at  a,  time  is  nl     Hence 

P.p\qlrl  =  n\        P=      .^;      . 

plqlrl 

By  a  similar  proof,  the  number  of  permutations  of  71  things  n  at 
a  time,  p  of  which  are  alike,  q  alike,  r  alike,  ,  ,  , ,  t  alike,  is  seen 
to  be 

p\q\r\...t\         ('»=i'  +  ?+'-  +  ---+0- 

Eor  example,  the  number  of  permutations  of  all  the  letters  of  the 

11! 
word  Mississippi  is  ^^  ^  ^  —  34650. 

EXERCISES. 

1.  In  how  many  different  ways  can  five  boys  stand  in  a  row  ? 

2.  How  many  numbers  of  six  digits  can  be  formed  by  using  the  numbers 
1,  2,  3,  4,  5,  6  ?     How  many  numbers  of  four  digits  ? 

3.  How  many  numbers  of  three  digits  (the  first  not  zero)  can  be  formed 
with  0,  1,  2,  8,  4,  5  ?    How  many  of  five  digits  ? 


90  BINOMIAL    THEOREM,  [Ch.  IX 

4.  Find  the  number  of  permutations  of  all  the  letters  of  the  word  animal ; 
of  the  word  relative.  In  how  many  permutations  do  the  vowels  and  conso- 
nants alternate  ?     How  many  of  the  latter  end  with  a  vowel  ? 

5.  How  many  numbers  less  than  1000  can  be  made  with  the  digits  0, 1,  2, 
3,  4,  5,  6  ?    How  many  with  the  digits  1,  2,  8,  4,  5,  6,  7  ? 

6.  In  how  many  ways  can  seven  boys  form  a  ring  ?   a  row  ? 

7.  How  many  sums  of  money  can  be  made  with  4  pennies,  3  dimes,  2  quar- 
ters, and  1  dollar  ? 

8.  Prove  that  n  +  ^Gr-^l  =  nCr +  1  +  ^nGr-{-nGr-l. 

9.  If„C5  =  „(7io,  find^Cg. 

10.  In  how  many  ways  can  4  men  and  2  boys  be  chosen  from  12  men  and 
5  boys  ?       -    '•  ^      C>C     } 

11.  In  how  many  ways  can  six  gentlemen  and  six  ladies  be  arranged  at  a 
round  table  so  that  no  two  gentlemen  are  adjacent  ? 

12.  How  many  triangles  can  be  formed  by  joining  three  of  the  vertices  of 
a  general  octagon  ? 

13.  Prove  that  nCr  +  \  =  n-lGr  +  n-^Gr  +  n~^Gr-\-  .  .  .  -\-rGr. 

14.  By  considering  the  combinations  of  s  letters  some  of  which  are  chosen 
from  rtj,  «2J  •  •  •  >  <hy  ^^^  t^^  others  from  h^,  b^,  .  .  .  ,  bq,  show  that 

p  +  q^s   =^  p^s   "I     pt/V—  1  •  qOi  -f-  pt's  —  2  •  q^2    r  •  •  •     \    P^l  •  q^s  —  1  -p  gC/s* 

BINOMIAL    THEOREM   FOR   POSITIVE    INTEGRAL   INDEX. 

91.  Let  71  be  a  positive  integer  and  consider  the  product 

{a  -\-  h){a  -{-  b) ,  ,  .  {a  +  b)  [n  factors]. 

One  term  of  the  product  is  a^;  it  is  obtained  by  taking  the  letter  a 
from  each  parenthesis.  There  will  be  n  terms  a'^~'^b,  since  the  let- 
ter b  may  be  chosen  from  any  one  of  the  7i  parentheses,  which  may 
be  done  in  ^(7,  =  n  ways.  There  will  be  ^0^  terms  a"'~^^,  since  two 
Fs  may  be  chosen  from  any  two  of  the  n  parentheses  and  the  a's 
from  the  remaining  parentheses.  In  general,  there  will  be  „C^ terms 
(in-r^r^  sincc  the  r  Fs  may  be  chosen  from  any  r  of  the  ri  parentheses, 
the  a's  being  then  taken  from  the  remaining  parentheses.     Hence 

(a  +  by  =  (Ot+^C.a^-'b+^C^a'^-'P  +  .  .  .  +  ^C^ -'^''+  •  . .  +  ^^ 

The  expression  on  the  right  is  called  the  expansion  of  {a  +  by\ 
and  the  theorem  involved  is  called  the  binomial  theorem  for  the  cato 


Sec   92]  COLLEGE  ALGEBRA,  9I 

in  which  the  index  z;^  is  a  positive   integer.     The  r  -\-  1st  term   is 
,^C ,xt'' " '^V \  it  is  called  the  general  term  of  the  expansion. 
Suhstitnting  for  ,,(7^,  ^Cj,  .  .  .  their  valnes  (§  87),  we  get 

1  *  2  ...  r 

^y  §  88,  ,,(7;,  =  „C',,_,,for  all  values  of  r  from  1  to  n.  Hence  in 
the  expansion  of  [a  -\-  by\  the  coejjicieiits  of  any  two  terniti  equidis- 
tant from  the  beginning  and  the  end  are  equal. 

92.  To  determine  the  greatest  coefficient,  we  observe  that 

__n(ii  —  \)..,{n—r-\-2){n  —  r^\) 
"^  ~  ^1  •  27rrr~-^  1  •  r  ' 

p        _  ^(^^  —  1) . .  .  {n  —  r  -\-  2) 
"^-'  ~  1  •  2  .  .  .  r  -  1  • 

n-~r  +  l 

>  ^i ^  _|_  X  > 

Hence  „(7,,  =  ,, (7^ _i  according  as =  1,  namely,  according 

as  r  —^(pi-\-l),  since  the  inequality  sign  must  be  reversed  when 

the  signs  of  the  two  members  are  changed.  Hence  the  number  ,,(7^ 
increases  as  r  increases,  so  long  as  r  is  less  than  ^{n  -\-  1),  but  de- 
creases as  r  increases  when  r  >  ^{71  -\- 1).  The  case  r  —  ^{71  +  1) 
occurs  only  when  n  is  odd;  then  ?•  —  1  =  ^  —  r,  so  that  ^0^  =  n^r-u 
in  agreement  with  the  result  of  §  88.  Hence,  if  71  is  odd,  the  coef- 
ficients n(^hoi  +  i)  ^^^^  n^h(n-i)  ^^'^  cqual  and  are  greater  than  the 
remaining  coefficients;  if  71  is  even,  the  coefficient  ^C^^  is  the  great- 
est coefficient. 


92  MULTINOMIAL    THEOREM,  [Ch.  IX 

Observe  that  the  number  of  terms  in  the  expansion  of  {a  +  ly  is 
n-^-l,  and  that,  for  n  even,  „Q„  is  the  coefficient  of  the  ^7i  +  1st 
term,  which  is  then  the  middle  term ;  while  for  7i  odd,  there  is  no 
middle  term,  but  there  are  two  terms,  the  ^{n  +  l)st  and  the 
\{n  +  3)rd,  at  the  middle  of  the  expansion,  their  coefficients  being 
n^i(»-i)  9,nd  n^i(n  +  i)'     ^c  may  state  the  result: 

In  the  expansion  of  {a  +  Vf,  the  greatest  coefficient  is  the  ..licldle 
term  if  n  he  even  ;  while  for  n  odd  the  middle  pair  of  terms  have  the 
greatest  coefficients, 

EXERCISES. 

1.  Find  the  greatest  coefficient  in  the  expansion  of  (a  -f  Vf, 

2.  Find  the  greatest  coefficients  in  the  expansion  of  {a  -j-  Hf. 

3.  Two  coefficients  „Cr  and  nCs  are  equal  only  when  r-\-8  =  n  or  r  =  8. 

4.  If  the  10th  and  12th  coefficients  are  equal,  find  the  4th. 
6.  If  2«C3  =  24nC4,  find  71. 

6.  The  sum  of  the  coefficients  in  the  expansion  of  {a  -f  &)^  is  2". 

7.  The  sum  of  the  coefficients  of  the  odd  terms  of  a  binomial  expansion 
equals  the  sum  of  the  coefficients  of  the  even  terms. 

8.  If  there  be  a  middle  term,  its  coefficient  will  be  emn. 

9.  nCi+2nC2  +  3nCs+.  •  •  -^  r  nCr   +.  .  .  +  7i  n  Gt  =  /i  2"-l. 

10.  nCx  -2n(7-,  +  3n6^3-  .  .  .  +  (  —  l)"-l^i  nO.  =  0. 

11.  nCo  +  inO«  +  Jn(74  +  i„a+...    -  ^. 

{2n)  ! 


12.    uOq  nOi  -j"  '^^i  nOi  -f-  n^a  n^3  -}-...  -{-nOn—1  nOn  — 


(w  4-  1) !  (71  -  1)  !* 


13.  nCx  -  i„(7u  +  inC3  -  .  .  .  +  (  -  1)^-^1  n(7n  =  i  +  4+  J  +  .   .  .  +  ^- 

14.  nCo«  +  nO.'  +  n^.'  +  .  .  .  +  nCn'  =  (2^)  !  -  (t^  !  )K 

MULTINOMIAL  THEOEEM. 
93.  Let  7^  be  a  positive  integer  and  consider  the  product 

(a+Z>  '\-c-{-d){a-{-h-\-c-\-d)  .  .   .  {a+I)-\-c+d)^      [n  factors]. 
Every  term  of  the  product  is  of  the  form 

each  exponent  a^  /3,  y,  S  being  a  positive  integer  or  zero.     But 
various  terms  will  be  equal  and  may  be  added.     For   a,  p,  y,  6 


Sec.  93]  COLLEGE  ALGEBRA.  93 

given,  the  terms  equal  to  a'b^c^d  may  all  be  obtained  by  selecting 
the  letter  a  from  a  parentheses,  h  from  (3  parentheses,  c  from  y 
parentheses,  and  d  from  the  remaining  6  parentheses,  the  selection 
to  be  made  in  every  possible  way.  By  §  90,  this  selection  may  be 
dene  in  exactly 

n\ 
a\  fi\y\d\ 

ways.     Hence  the  general  term  in  the  expansion  of  (rt+J+c-j-^)"^  is 

n  ! 


a  \  fi  \  y  \  d 


-^a%^c''d\ 


Example  1.  Find  the  coefficient  of  a^h'^c'^  in  (<z  +  6  -f  cf 

7  ! 
The  coefficient  is  „  =  210. 

Example  2.  Find  the  coefficient  of  x^  in  {a  -{-  bx  -{-  cx'^f. 
The  general  term  of  the  expansion  is 

a\ p\y  ! 

We  seek  the  terms  in  which  ft  -\- 2y  —  5.     Each  of  the  quantities  a,  /?,  y 
is  to  be  a  positive  integer  or  zero.     Hence  x  <  2-     ^^^  ;^  =  2,  we  get  ft  =  1, 

or  =  5  ;  for  ;k  =  1,  /i  ^  3,   a  =  4:  ;  for  y  =  0,  ft  =  6,  a  =  d.     The  coefficient 
of  x^  is  therefore 

8  '  8  '  8  ' 


5 !2 !  '    4!3 !  '     3!5! 


EXERCISES. 


1.  Find  the  coefficients  of  a^^c  and  a^b  in  (a  -j-  &  -f  c)^ 

2.  Find  the  coefficient  of  a'^b^c*d  in  (a  +  6  +  c  +  d)^^. 

3.  Find  the  coefficient  of  x^  in  (1  -f-  3jj  -f-  4a;  ^y. 

4.  Find  the  coefficient  of  (c**  in  (1  -}-3a;  —  x'^f. 

5.  Find  the  coefficient  of  x*  in  (1  +  2x  -j-dx''  -f  4ic')^ 


CHAPTER  X. 

PROBABILITY  (CHANCE). 

94.  Definitions.  If  an  event  can  happen  in  m  wa3^s  and  fail  in 
n  ways,   and  if  each  of  these  m  -\-  n  ways  is  equally  likely,  the 

probability,  or  the  chance,  of  its   happening   is  - — ~p — ,  and  tlie 

probability  of  its  failing  is  — ; — . 
^  ''  ^      7H  -\-  n 

An  equivalent  statement  is  that  the  odds  are  m.  to  n  in  favor  of 
the  event,  or  n  to  m  against  the  event. 

For  example,  if  a  single  ball  is  drawn  at  random  from  a  bag 
containing  4  white  and  6  black  balls,  the  chance  of  drawing  a  white 
ball  is  3^Q-  and  the  chance  of  drawing  a  black  ball  is  y-g-.  Tlie 
chance  that  the  ball  will  be  either  white  or  black  is  y\  -f-  y'^-Q  :r=  1 , 
iijimcly,  one  of  the  events  is  certain  to  happen.  The  probability  1 
denotes  certainty. 

95.  Example  1.  From  a  bag  containiug  4  white  and  G  black  balls,  2  balls 
are  drawn.  Find  the  chance  d)  that  both  are  white  ;  (2)  that  both  are  black  ; 
(3)  that  one  is  white  and  one  black. 

The  total  number  of  ways  in  which  2  balls  can  be  drawn  from  10  is 
.pTo  —  45.  The  number  of  ways  of  drawing  2  white  balls  is  4 ^'2  =  6; 
of  drawing  2  black  balls  oCa  =  15  ;  of  drawing  1  white  and  1  black  ball 
J\  ,  f,C\  —  24.  As  a  check,  we  note  that  6  -f  15  -f  24  —  45.  The  chances  of 
the  respective  events  are  therefore  /-,  |f ,  f  J. 

Example  2.  A  man  has  3  chances  in  a  lottery  in  which  there  are  5  prizes 
and  10  blanks.     Find  his  chance  of  drawing  a  prize. 

Of  the  15  tickets,  3  may  be  drawn  in  ,6(7:,  —  455  ways.  3  prize  tickets 
may  be  drawn  in  oCa  =  10  ways  ;  2  prize  tickets  and  1  blank  may  be  drawn 

94 


Sec.  95] 


COLLEGE  ALGEBRA, 


95 


in  sCa  .  \qCx  ~  100  ways  ;  1  prize  ticket  and  2  blanks  in  SioCa  —  225  ways  •, 
3  blanks  in  106^3  =  120  ways.     The  respeqtive  chances  are 

a.-9  10  «    100       .  .  -  22  5         .120 

whose  sum  is  1,  a  check  upon  the  work.  The  chance  of  drawing  one  or  more 
prizes  is  the  sum  of  the  first  three  chances,  viz.,  f||. 

Example  3.  If  5  coins  are  tossed,  what  is  the  probability  that  3  heads  and 
2  tails  will  appear  ? 

As  each  coin  may  fall  in  2  ways,  the  5  coins  may  faL  in  2'  =  32  ways. 

These  events  may  be  classified  as  follows  :  5  heads  may  fall  in  1  way  ;  4 

heads  and  1  tail  in  5  ways  ;  3  heads  and  2  tails  in  10  ways ;  2  heads  and  3 

j  tails  in  10  ways  ;  1  head  and  4  tails  in  5  ways  ;  5  tails  in  1  way.    As  a  check, 

j  we  observe  that  1  -|-  5  +  10  +  10  +  5  +  1  =  32.     The  chance  for  3  heads 

j  and  2  tails  is  therefore  ^.5. 

Example  4.  If  a  pair  of  dice  are  thrown,  find  the  probability  that  the 
sum  of  the  face  number-;  will  be  6. 

As  each  die  has  6  faces,  the  pair  can  be  thrown  in  6  X  6  =  36  different 
ways.  The  desired  events  are  of  three  kinds  :  1  and  5,  2  and  4,  3  and  3. 
Now  the  first  die  may  come  up  1  and  the  second  5,  or  mce  versa,  so  that  the 
pair  may  be  1  and  5  in  two  ways.  Similarly,  the  pair  may  be  2  and  4  in  two 
ways.  The  paii  can  be  8  and  3  in  a  single  way.  Hence  the  probability  of 
throwing  6  is  -f^  +  i\  +  3V  =  i^. 

Example  5.  In  a  lottery  there  are  10  blanks  and  3  prize  tickets  of  $100 
each  and  2  prize  tickets  of  $10  each.  What  is  the  value  of  the  expectation  of 
a  holder  of  3  tickets  ? 

We  form  a  table  in  which  the  first  column  indicates  the  nature  of  the 
event,  the  second  column  its  probability,  the  third  column  the  total  value  of 
the  event  if  successful,  the  fourth  column  the  value  of  the  expectation  (prob- 
able value)  of  the  event. 


Event. 


Three  $100  prizes 

Two  $100,  one  $10 

One  $100,  two  $10 

Two  $100,  one  blank  .... 

Two  $10,  one  blank 

One  $100,  one  $10,  one  blank. 

One  $100,  two  blanks 

One  $10,  two  blanks 

Three  blanks 


Probability. 

Value  of  Event. 

Expectation. 

1  --455 

$300 

$300  -f-  455 

6-455 

210 

1260  -^  455 

3  -^  455 

120 

360  -f-  455 

30  ^  455 

200 

6000  --  455 

10  -^  455 

20 

200  ^  455 

60  -^-  455 

110 

6600  ^  455 

135^455 

100 

13500  --  455 

90  -^  455 

10 

900  -^  455 

120  -^  455 

0 

0 

The  sum  of  the  numbers  in  the  last  column  is  29120  -^  455,  or  64.  Hence 
the  expectation  of  the  holder  has  the  value  $64.  As  a  check,  the  sum  of  the 
fractions  in  the  second  column  is  unity. 


9^  PROBABILITY  {CHANCE),  [Ch.  X 

96.  If  ttvo  events  are  independent  *  (the  occurrence  of  one  not 
contingent  upon  the  occurrence  of  the  other) ^  the  prohahility  that  hoth 
ivill  happen  equals  the  prohahility  of  the  first  event's  happening 
multiplied  by  the  prohahility  of  the  second  evenVs  happening.  I 

Suppose  that  the  first  event  can  happen  in  m  ways  and  fail  in  n  \ 
ways,  each  of  these  ways  being  equally  likely  to  occur.     Let  the  I 
corresponding  numbers  for  the  second  event  be  if  and  N.    We  may  j 
associate  any  one  of  the  ni  +  n  occasions  in  which  the  first  event 
happens  or  fails  with  any  one  of  the  M  -\-  N  occasions  for  the  second 
event.       Consider  the  resulting  (pi  -j-  n)(M -\-  N)  occasions.     In 
mM  oi  these  occasions  both  events  happen.     Hence  the  probability 
that  both  events  will  happen  is 

^^  Prob.  of  1st  X  Prob.  of  2d. 


{m  +  n){M+  JV) 

In  general,  it  P^,  F^,  P^,  .  , .  are  the  respective  probabilities  of 
a  number  of  independent  events,  P^-  P^-  P^.  . ,  is  the  probability 
that  all  will  happen. 

For  example,  the  probability  of  throwing  an  ace  in  the  first  and| 
second,  but  not  in  the  third,  of  three  successive  throws  of  a  die  is 
T  >^  i  X  f  =  2"f6''  since  the  probability  of  throwing  an  ace  at  one 
throw  is  ^,  and  the  probability  of  not  throwing  an  ace  is  f . 

Example  (Poincare).  From  two  urns  of  like  exterior,  the  first  contain-  \ 
ing  4  white  and  6  black  balls,  the  second  containing  3  white  and  2  black  balls^ 
one  ball  is  drawn  at  random.  Find  the  probability  P  that  a  white  ball  is  I 
drawn. 

Since  the  total  number  of  balls  in  the  two  urns  is  15  and  the  number  of; 
white  balls  is  7,  one  might  be  led  to  say  that  the  probability  P  of  drawing  a 
white  ball  is  7/15.  But  this  solution  is  incorrect  as  it  assumes  that  the  events 
are  equally  probable.     Indeed,  the  x)robability  of  drawing  a  particular  ball  I 

*We  may  state  the  condition  that  ^  and  B  are  independent  events  in  the 
following  form.     The  probability  that  A  will  happen,  the  probability  that  A  \ 
will  happen  if  P  happens,  and  the  probability  that  A  will  happen  if  B  does  not  i 
happen  shall  all  three  be  equal.      See  Poincare,  Calcul  des  Probabilites, 
Paris,  1896. 


Sec.  96]  COLLEGE  ALGEBRA,  97 

from  the  first  urn  is  1/10,  while  the  probability  of  drawing  a  given  ball  from 
;the  second  urn  is  1/5. 

The  correct  solution  is  as  follows.  As  each  urn  is  equally  likely  to  be 
selected,  the  probability  that  the  first  urn  will  be  selected  is  \,  the  probability 
that  the  second  urn  will  be  selected  is  |.  If  a  ball  is  taken  from  the  first  urn, 
ithe  probability  that  it  will  be  a  white  ball  is  -^q.  Applying  the  theorem 
which  relates  to  two  independent  events,  we  find  that  the  probability  that 
ithe  first  urn  will  be  selected  and  that  a  white  ball  will  be  taken  from  it  is 
=  Y%.  Analogously,  the  probability  that  the  second  urn  will  be  selected 
"and  that  a  white  ball  will  be  taken  from  it  is  J  .  |  =  }^.  The  required  prob- 
ability P  is  therefore  f^  -j-  ^^^  =  ^. 

EXERCISES. 

1.  From  a  bag  containing  4  white  and  8  black  balls,  3  balls  are  drawn. 
•Find  the  chance  (1)  that  all  are  white  ;  (2)  that  2  are  white  and  1  is  black. 

2.  A  has  3  shares  in  a  lottery  in  which  there  are  3  prizes  and  15  blanks  ; 
B  has  2  shares  in  a  lottery  in  which  there  are  3  prizes  and  9  blanks.  Com- 
pare their  chances  of  drawing  at  least  one  prize. 

3.  If  6  coins  are  tossed,  find  the  chance  for  3  heads  and  3  tails. 

4.  If  3  Latin,  5  Greek,  and  6  French  books  are  placed  on  a  shelf  at  random, 
find  the  chance  that  the  books  of  each  language  will  be  together. 

5.  The  letters  of  the  word  probability  are  placed  at  random  in  a  line. 
Find  the  chance  that  two  vowels  will  come  together. 

6.  Find  the  chance  for  a  sum  4  at  one  throw  of  two  dice. 

7.  Find  the  chance  for  a  sum  15  at  one  throw  of  three  dice. 

8.  Find  the  chance  for  a  sum  of  at  least  6  at  one  throw  of  two  dice. 

9.  In  a  lottery  of  100  tickets,  there  are  5  prizes  of  |100,  10  of  |50,  and  10 
of  $20.     Find  the  value  of  a  ticket. 

10.  From  a  bag  containing  5  twenty-dollar  bills  and  15  ten-dollar  bills,  a 
person  is  entitled  to  draw  2  bills.     What  is  the  value  of  his  expectation  ? 

11.  Find  the  probability  of  throwing  an  ace  at  least  once  in  three  succes- 
sive throws  of  a  single  die. 

12.  From  a  bag  containing  4  white,  5  black,  and  6  red  balls,  3  balls  are 
drawn  in  succession,  each  being  replaced  prior  to  the  next  draw.  Find  the 
probability  that  (1)  the  balls  are  of  different  colors  ;  (2)  the  first  is  white,  the 
second  black,  the  third  red  ? 

13.  If  two  events  are  dependent  (the  occurrence  of  one  being  contingent 
upon  the  occurrence  of  the  other),  the  probability  that  both  will  happen  equals 
the  probability  of  the  first  multiplied  by  the  probability  that  when  the  first 
has  happened  the  second  will  follow.  Prove  a  similar  theorem  for  any  num- 
ber of  events. 

14.  Solve  Ex.  12  when  the  balls  are  not  replaced  after  being  drawn. 


98  PROBABILITY  {CHANCE).  [Cii.  : 

15.  If  Pis  the  probability  of -the  happening  of  an  event  and  therefor 
1  —  P  the  probability  of  its  not  happening,  the  probability  that  the  event  wil 
liappen  exactly  r  times  in  n  trials  is 

nCr  P'Xl-P)^-^ 

16.  If  6  coins  are  tossed,  find  the  chance  for  a  single  head. 

17.  Find  the  chance  of  throwing  exactly  3  aces  in  6  throws  of  a  die. 

18.  Find  the  chance  of  throwing  at  least  3  aces  in  6  throws  of  a  die. 

19.  The  probability  that  A  can  solve  a  given  problem  is  ~  and  B's  proba 
bility  is  f .     Find  the  probability  that  both  will  solve  it. 

20.  A  number  of  3  digits  is  formed  at  random  from  the  10  figures  0,  1 
2, . . . ,  9.     Find  the  chance  that  the  sum  of  the  3  digits  is  25. 

21.  A's  probability  of  being  alive  20  years  hence  is  J,  B's  probability  is  | 
Find  the  probability  that  both  will  be  alive. 

22.  Two  balls  are  drawn  from  a  bag  containing  3  red,  4  white,  and  5  blac 
balls.     What  is  the  chance  that  both  are  of  the  same  color  ? 

23.  A  has  twice  the  skill  that  B  has.  What  is  the  chance  that  A  wins  ' 
games  before  B  wins  3  ? 

24.  What  is  the  chance  that  in  5  numbers  taken  at  random  exactly  tw^ 
begin  with  the  digit  9  ?    That  at  least  two  begin  with  9  ? 


CHAPTER   XL 

MATHEMATICAL  INDUCTION. 

97.  By  the  method  of  snraming  an  arithmetical  progression  we 
find  that  the  sum  of  the  first  ^z- odd  numbers  is '?^^.  This  result 
may  also  be  proved  by  tlie  following  method.  We  observe  that 
1  +  3  =  2'^  1  +  3  +  5  r=  32,  1  +  3  +  5  +  7  =  42.  Let  us  suppose 
that  we  have  continued  this  numerical  verification  of  the  theorem 
as  far  as  the  first  m  odd  numbers,  so  that 

1  4-  3  _j_  5  4-  .  .  .  -f  (2m  -  1)  =  ?7^l 

Consider  next  the  sum  of  the  first  m  +  1  odd  numbers.  This  sum 
may  be  derived  from  the  preceding  sum  by  adding  the  m  -\-  1st  odd 
number,  which  is  2m  +  1.  Adding  it  also  to  the  second  member 
m^,  we  get 

1_}_34_5  +  .  .  .  +  (2w-l)  +  (2m+l)=rm2+(2m+l)  e  (ni+lf. 

In  the  left  member  of  this  equation  we  have  the  sum  of  the  first 
m-\-l  odd  numbers  ;  in  the  right  member  we  have  the  square  of 
m  -\-  1.  Hence  the  theorem  holds  also  for  n  =  ??i  -f-  1  when  it  has 
'been  established  for  n  =  m.  But  we  observed  from  the  identity 
1  -|-  3  =  2^  that  the  theorem  holds  for  ?i  =  2  ;  hence  b}  our  proof 
it  holds  for  7^  =:  2  + 1  =  3.  Being,  true  for  n  =  3,  the  theorem 
holds  for  7^  =  3  +  1  =  4.  Being  true  for  n  =  4,'  it  holds  for 
71  =  4  -[-  1  =  5,  «tc.  It  therefore  follows  that  the  theorem  holds 
for  any  positive  integer  n. 

99 


loo  MATHEMATICAL   INDUCTION,  [Ch.  XI 

98.  This  method  of  proof  is  called  mathematical  induction. 
There  are  two  distinct  parts  in  the  proof  as  applied  to  establish  a 
theorem  true  for  every  positive  integral  value  of  n.  On  the  one 
hand,  we  have  to  show  that,  if  the  theorem  holds  for  any  particular 
value  m  of  7i,  it  will  then  hold  for  the  next  value  m  -\- 1  oi  n.  On 
the  other  hand,  we  have  to  show  that  the  theorem  is  true  for 
the  initial  values  of  n,  say  ^  =  1,  or  ^^  =  2,  so  that  there  may  be  a 
starting  point  (free  from  '^ifs^')  for  the  induction  process  from  m 
to  m  -\-\  (from  '^  if  true  f or  m  "  to  '^  then  true  for  m  +  1 ").  We 
must  have  a  ladder  by  which  to  climb  from  any  round  (the  mth)  to 
the  next  round  (the  m  +  1st) ;  but  the  ladder  must  rest  on  a  solid 
basis  so  that  we  'can  get  on  to  the  ladder  (the  ?^  =  1  or  ^  =  2 
rounds). 

To  illustrate  the  necessity  of  both  parts  in  the  proof,  consider 
the  two  following  examples. 

Whatever  be  the  value  of  the  number  c^  the  equation 

1  +  2 +  22  + 23  +  ...  +  2"  =  2"  +  ^  +  c 
will  be  true  for  ?i  =  m  +  1  if  true  for  n  =  m.     Indeed,  we  have 
(l_^2  +  2-^  +  ...+2"^)  +  2"^  +  i=  (2^  +  ^+c)  +  2"*  +  izz:2^  +  2_^c, 

which  is  the  original  formula  for  ^  =  m  +  1.  The  ^^ ladder"  is 
therefore  perfect.  But  to  be  able  to  climb  up  the  ladder  to  the 
general  round  (the  ^th  round),  we  must  be  able  to  get  on  the  first 
round  ;  the  formula  must  be  true  for  n  =  1,  which  requires 
1  +  2  =  2^  +  c,  whence  c  =  —  1.  *   Hence  the  only  true  formula  is 

1  +  2  +  2'-^  +  2'^+  .  . .  +  2^  =  2"  +  ^  -  1,  I 

a  result  also  proved  by  the  formula  for  the  sum  of  a  geometrical 
progression. 

As  the  second  example,  consider  the  theorem  due  to  Gauss 
that,  for  prime*  numbers  n,  the   circumference  of  a  circle  can 

*  A  positive  integer  n  is  called  prime  when  it  is  divisible  by  no  positive 
integers  other  than  unity  and  n  itself.     Thus  2,  3,  5,  7,  11, ... ,  are  primes. 


Sec.  99]  COLLEGE  ALGEBRAl   '  loi 

be  divided  into  ii  equal  parts  by  means  of  ruler  and  compass  if,  and 

only  if,  71  has  the  form  7^  =  2^""  +  1.  For  m  =  0,  1,  2,  3,  4,  we 
find  that  n  —  3,  5,  17,  257,  65537,  respectively,  and  that  each  of 
these  values  of  7^  is  a  prime  number.  It  was  consequently  supposed 
by  Fermat  that  the  expression  22*^+1  was  a  prime  number  for  all 
positive  integral  values  of  m.  Later,  however,  Euler  proved  that 
this  supposition  was  false  by  showing  that,  for  m  =  5,  the  resulting 
number  2'^'  +  1  =  2^2  +  1  ==  4,294,967,297  is  divisible  by  641. 

99.  Example  1.  Prove  by  mathematical  induction  that  the  sum  of  the 
cubes  of  the  first  n  natural  numbers  equals  \ln{n  -\- 1) p. 

By  trial,  1^  +  2^  ==  { ^2(3) }  ^    Suppose,  then,  that  the  theorem  holds  for  any 
particular  value  m  of  n,  so  that 
1^:  lH-2'+33-f-...+m3  =  ||w(m  +  l)}2. 

Adding  {m-\-lf  to  both  members,  we  get 

13  _|_  23  4-  33  -f  .  .  .  -I-  m^  +  (w  +  If  =.  (7/1  -f  1)^  I  (T^j  +  1)  +  ^'  [ 

=  (^  +  l)^{^^^^[={i(m4-l){-  +  2)l^ 

\  Hence  the  theorem  holds  also  for  the  value  m  -\-lof  n.  Being  true  for  71  =  2, 
the  theorem  therefore  holds  for  n  —  Z,  etc.  By  the  two  parts,  the  theorem  is 
proved  to  hold  for  every  n. 

Example  2.  Show  that  x^  —  y^  is  divisible  by  a?  —  y  if  71  is  a  positive 
I  integer. 
I       The  theorem  is  evidently  true  for  n  —  i  and  /i  =  2.     Since 

ajj.t  —  ym^x'^-^{x  —  y)-\-  y{x^  ~  ^  —  2/"*  ~  ^)» 

the  theorem  will  be  true  for  n  =:  m  \i  true  for  n  —  m—\.  In  fact,  if 
a;"*  - 1  —  2/"^  - 1  is  divisible  by  oj  —  y,  the  identity  shows  that  a;"*  —  y"^  will  also 
ibe  divisible  by  a?  —  y.     Hence  the  theorem  follows  by  induction. 

Example  3.  Prove  by  induction  the  Binomial  Theorem  for  all  positive 
integral  exponents  n. 

By  §  91,  the  theorem  in  question  is  given  by  the  formula 

{a  +  hf'  =  c^a^  +  c^a^-^b  -f-  c^a^-'^b"^  + .;. .  -f  Cra^-^br  + . . .  +  <j„6» 
where 

^  71(71  —  1)  n(n  —  l)..,(n  —  r-\-l)  ^ 

Co  =  1,  Ci  =  71,  C2  =  -\t2-'  '"><^r=  ~ 1'^     _r  '  .  .  .  ,  tJn  =  t. 


lo?  MATHEMATICAL  INDUCTION.  [Cii.  X] 

The  theorem  is  true  for  n  =  1,  since  {a -\- h)^  —  a -{- b  ;  also  for  tz  r=  2,  since 

{a  +  bf  =:  d-^  +  2ab  +  6'^  r^  Co^^  -f  c^ah  -f  Cg^'^.  J 

Suppose  that  the  theorem  has  I)een  established  for  7i  —  m,  so  that 

Multiplying  both  members  by  a  -f  h,  we  get 

(a  +  bf^  + 1  =  ro^^"  ^ ^  +  (^0  +  c^)a'^b  +  (c^  +  c^'ia'^-^b^  +  .  .  . 

But  ^0  =  1,  Cm=  1 ,  and 

Co  +  ^1  =  ^  4-  1, 

m{m  —  1)  _  'm,{m  +  ^) 


Ci  +  ^2  =   ^  +  ■ 


1-2         "        1-2 


_  w(77i  —  1) . .  .  (w  —  r  --f-  2)      ??^(m  —  1) .  ,  .  (7?^  —  r  -f  1) 
c^_i  +  Cr  -  1^2^77(7^1)  ^  1-2.  ..r 

m[m  —  1) .  . .  (m  —  r  +  2)[r  +  (?7J  —  r  +  1)| 
~  ~  1  -2.  ..(r  -  l)r  ' 

(?y^  +  l)7/^(m  —  1) ...  (m  —  r  -f  2) 
"^  1  •  2  .  .  .  (r^^  1)7- 

..,       (a-^^>)»n  +  l  :=«ni  +  l   +  (m  +  1)^^^  +    '^^  ^  ^^^^  ^"^  "  ''^'  +  '  '  * 

But  this  formula  is  the  same  as  the  initial  formula  for  {a-\-by^  ior  n  =  m-\-l. 
Hence  the  theorem  is  true  for  n  =  m-\  1  if  true  for  n  =  m.  Also  the  theoreii 
is  true  for  7i  =  1.  Hence  it  is  always  true.  In  fact,  being  true  for  7i  =  1,  ii 
is  true  for  n  —  2,  and  therefore  for  n  =d,  etc. 

Example  4.  If  n  is  a  prime  number,  iV^"—  iV^is  divisible  by  n. 

Denote  iV"  —  iV^by  the  functional  symbol /(iV^),     Then 

/(Jf+l)  -f{M)  =  \(M-{- 1)«  -  (1/+  1;}  -  {.¥«  -  M] 

1   /w 
upon  expanding  {M-\-  1^  by  the  Binomial  Theorem.    The  first  and  last  terms 
are  evidently  divisible  by  n.     Also  is  an    integer,   bcinLr  a  binomia" 

coefficient,  and  is  divisible  by  n,  since  2  does  not  divide  tlie  i  rime  n  {n  >  2, 
otherwise   the    term   M'^--  does   not   occur).       In   geucral,    the   coefficienl 

^'^~    1  .V  ' —  ^^  -¥«-»*  occurs  only  when  n  >  r  and  is  then 


Sp:c.  99J  COLLEGE  ALGEBRA.  103 

mteger.  Moreover,  it  is  divisible  by  n  since  there  is  no  factor  in  coninion 
witli  n  and  thc3  denominator  r !  In  fact  n  is  greater  than  r  and  hence  cannot 
divide  any  factor  of  r  ! ;  while,  inversely,  no  factor  of  r  !  can  divide  the  prime 
number  n, 

.'.    /(i/+ 1)  ^f{M)  -f  a  multiple  of  n. 

Hence,  if  /  {M ,  is  divisible  by  n,  so  is  also/(J^f  +  1).  But/(1)  =  0.  Hence 
f{2)  is  divisible  by  n  ;  therefore  also/(3),  etc. 

This  theorem  U  known  as  Fermat's  Theorem.  Incidentally  it  was  shown 
that  in  the  expansion  of  {a  -f  &)^  for  n  a  prime  number,  all  the  binomial  coef- 
ficients except  the  first  and  last  are  divisible  by  n. 

As  illustrations  of  the  theorem,  we  observe  that 

2^  -  2  =z  5  X  6,     8^  -  3  =  5  X  48,     4^^  -  4  =  5  X  12  X  17. 

EXERCISES. 

Prove  by  mathematical  induction  that 

1.  P  4-  2-'  +  32   h  .  .  .  +  ^'  ^  ln{n  +  l)(2/i  +  1). 

1-2  "^2  3"^  3.4  ■"^*  •  •  '^  n{n  i-l)~  n  -\-l' 

3.  {a  -^  b  -\-  c  -{-  dy^  —  a^^  —  b^  —  c^  —  d"  is  divisible  by  the  prime  n. 

4.  x^  —  y^^  is  divisible  by  ic  -|-  ^  when  n  is  even 

6.  2.4 -f 4.6  +  6-8  +  ... +27^(272,  + 2)  =rz^-(2ri  +2X2/1  +  4). 


^    ^   ,  m     '  m{m  +  1)   ,  m{7n  +  l)(m  +  2)  ,         ^         ,    .  ^  . 

7.  1  +  J  +  ""i7^  +  —  1 : 2  .  3  +  . .  .  to  7i  +  1  terms  equal** 


6.  2  7'^  +  3-o'»  —  5  is  divisible  by  24. 
1)  m(?7^  +  l)(m 
"~  "^  1  •2-3 

(m  4-l)(m  +  2).  .  .  (m  +  n) 

r^2 . . .  w 

8.  l-2  +  2-3  +  3-4  +  ...+  7i(7i  +  l)  =ri7^(7l  +  l)(^l  +  2). 

9.  l-2-3  +  2-3-4  +  ...+7i(/i  +  l)(?i  +  2)  =  ln{n  +  1 K^  +  2)(7i  +  3)^ 

10.  1=^+2=^+  . . .  +  ^^'  =  (1  +  2  +  . . .  +  71)'^. 

11.  (1^  +  25  +  .  .  .  +  7l5)  +  (17  4-  2^  +  .  .  .  +  71^)  :=r  2(1  +  2  +  .  .  .   f  '/^^ 

12.  cl'  +  2^  + . . .  +  ?i.^)  +  3(1^  +  2^  +  . . .  +  71"^)  =  4(1  +  2  +  . . .  -f-  /i)3. 


CHAPTER  XIJ 
LIMITS  ;  TNDETEKMWATE  FORMS. 

100.  Preliminary  Remarks  on  Limits. — In  testing  the  penetra- 
ting power  of  a  bullet  fired  at  a  given  distance  with  a  particular 
rifle  and  uniform  loads,  we  determine  the  limit  to  the  thickness  of 
the  pine  board  through  which  the  bullet  will  pass.  In  Geometry, 
the  length  of  the  circumference  of  a  circle  is  obtained  as  the  limit 
of  the  perimeters  of  the  inscribed  (also  of  the  circumscribed)  regular 
polygons  as  the  number  of  sides  increases  without  bound.  Of  all 
triangles  having  a  given  perimeter  and  a  given  base,  the  isosceles 
has  greatest  area,  so  that  there  is  a  limit  to  the  area  of  such  a  tri- 
angle. The  last  two  illustrations  offer  a  contrast  in  one  respect, 
the  variable  inscribed  regular  polygon  does  not  reach  its  limit,  the 
circumference;  but  in  the  last  example  the  variable  triangle  reaches 
the  limiting  form  of  greatest  area,  the  isosceles  triangle. 

In  the  algebraic  formulation,  we  consider  a  variable  x  and  a 
lunction/(^)  to  which  there  is  a  definite  value  for  each  value  that 
X  may  take  (see  definition  of  function,  §  75). 

A  Necessary  Condition  for  a  Limit. — In  order  that  the  func- 
i'ow  f{x)  shall  have  the  limit  I  as  x  approaches  a,  where  /  and  a 
are  fixed  quantities,  it  must  be  possible  to  make  the  difference 
between /(cc)  and  I  as  small  as  we  please  by  taking  x  sufficiently 
near  to  a,      [Complete  definition  of  limit  in  §§  103-104.] 

This  condition  is  satisfied  by  the  function  2^  -f  4  if  a  =  ^, 
^  =  10,  since  the  difference  between  2a;  -j-  4  and  10  is  proportional 

T04 


Sec.  101]  COLLEGE  ALGEBRA,  105 

to  the  difference  between  x  and  3.  In  this  case  we  may  let  x 
become  3,  whence  22:  +  ^  becomes  10. 

The  question  is  not  so  simple  with  the  function 

^  ^  ^       X  ^  a 

which  is  not  defined  for  x  =  a.  Indeed,  upon  setting  a;  =  «,  we 
are  led  to  the  symbol  ^ ,  which  may  be  regarded  to  be  equal  to  any 
number  whatever  if,  indeed,  it  have  a  value  at  all  (§  102).  At  any 
rate,  f{x)  does  not  have  a  definite  value  for  x  =  a  and  hence,  by 
the  definition  of  a  function,  is  not  defined  for  x  =  a.  We  may, 
however,  show  that  f{x)  approaches  a  limit  as  x  approaches  a.  As 
long  as  X  differs  from  a,  f{x)  equals  x  -{-  a.     Hence 

f{a  +  .01)  =  2a+  .01,         f{a  +  .001)  =  2a  4-  .001, 
f{a  +  .0001)  =  2a  +  .0001,  .  .  . 

By  making  the  difference  between  x  and  a  sufficiently  small, 
but  distinct  from  zero,  the  difference  between  f{x)  and  2a  may  be 
made  as  small  as  we  please.  That  this  necessary  condition  is  also  a 
sufficient  condition  for  the  limit  2a  is  shown  in  §  105. 

101.  The  Term  Infinity.— The  value  of  the  function  — for  any 

X 

value  of  X  except  zero  is  readily  computed.  The  operation  division 
ceases  to  have  a  meaning  when  the  divisor  is  zero  ;  there  is  no 
number  q  such  that  0  ,  q  =  10,  since,  for  every  number  q,  0 .  q  is 
always  0  ;    10    cannot  be  separated  into  parts   each  equal   zero. 

But  as  X  decreases  through  positive  values,  —  increases  and  may 

X 

be   made  to  exceed  any  assigned  value  (say  10,000)  by  taking  x 

sufficiently  small  (viz.,  less  than  one- thousandth).  Thus  — •  in- 
creases without  bound;  for  brevity,  it  is  said  to  ^^  become  infinite," 
or  to  '^approach  infinity^'   (symbol,  -[- 00 ,  or  merely  00),   terms 


► 


lo6  LIMITS;    INDETERMINATE  FORMS.  [Ch.  XII 

which    mean   only   *'  increases    without   bound. '^     Similarly,  as  x 
increases  through  negative  values,  —  decreases,  and  2i^x  approaches 

X 

zero,  —  decreases  without  bound  ;  it  is  said  to  become  negative 

X 

infinity  (symbol,  —  co  ).     A  variable  which  does  not  increase  with- 
out bound  nor  decrease  without  bound  is  said  to  remain  finite. 

\0x 
102.  While  tlie  function  —  is  not  defined  for  x—0,  division 

x 

by  zero  being  excluded  (§  101),  it  has  the  limit  10  as  x  approaches 

10a; 
zero.     In  fact,  —   =10  for  every  Xy^  0.     But  if  we  consider  the 

numerator   and   denominator  separately,  we  have  functions  which 

approach  0   as  a;  ^0^  and  the  quotient  of  their   limits  takes  the 

form    ^.       But   the   same   symbol   would   be   obtained    from    the 

%     ^x    \()x 
functions  — ,   --, ,  whereas  their  limits  as  a;  li:  0  are  1,  3,  10, 

XXX 

respectively.    Hence  the  symbol  ^,  considered  apart  from  its  origin, 

does  not  possess  a  definite  value  and  so  is  called  an  indeterminate 

form. 

lOx 
For  every  value  of  a;  >  0  the  function  —  has  the   value  10, 

so  that  its  limit  as  *  i?;  =  oc  is  10.  If  we  consider  the  numer- 
ator and  denominator  separately,  we  have  functions  which  increase 
without  bound  as  x  does,  and  the  quotient  of  their  limits  has  the 

form   ^.       But,  as  x  r:^  co ,  the  functions    -,    — ,    — -   have   the 

^  XX         X 

limits  1,  3,  10,  respectively,  while  each  may  lead,  in  the  sense  ex- 
f)Uiined,  to  the  symbol  ^-.  Hence  ^o  ^s  called  an  indeterminate 
form,  not  possessing  a  definite  value. 


*  Eead    "x  increases   without   bound."     x  ^  —  cc   is   read   '\v  dctreases 
without  bound."'    But  a'  =  1  is  read  **as  x  approaches  1." 


Sec.  102J  COLLEGE  ALGEBRA,  107 

Writing  the  above  functions  in  the  forms  x  ,  —^    x ,  -,    x  ,  — , 

vC/  %4y  JU 

and  considering  the  limits  of  the  separate  factors  as  x^^,  we  are 
led  to  the  symbol  0  .  go  in  each  case.     Similarly,  as  x  =  qo  ,  we  are 

X    S^    1 0^ 
led  to  the  symbol  00  .  0  in  each  case.      But  — ,  — ,  —  have  the  re- 
spective limits  1,  3^  10  as  :i^  — 0  or  as  a;  ==  go  .    Hence  tlie  symbols 
0  .  CO  and  co  .  0,  when  considered  apart  from  their  origin,  are  Mide- 
terminate  forms. 

As  x~l,  the  two  members  of  the  identity 

2 !_  __  1  -  x 


lead  to  the  respective  symbols  go  —  oo  and  ^.     Thus  go  —  00  is  an 
indeterminate  form,  when  considered  apart  from  its  origin. 

If,  for  X  ~  a,  di  function  of  x  assumes  an  indeterminate  form, 
we  determine  its  limit  as  ^  =  «.     If  the  given  function  \^f(x)/g{x), 

where  fix)  =  0  and  qix)  =  0  as  a;  =  «,  we  seek  -\A- 

•^  ^  x  —  a  g{x) 

Example  1.  As  x  ~  1,  (1  —  ^)/{\  —  ^^)  leads  to  the  symbol  0/0.     But 

1-x^  _  {i-x){l  -\-x-\-x^  -\-x^-\-ai^) 
l-a^~  [\  -  x){\ -{- X  ^Id^)         ' 

•.  limit      1  —  x^  _^ 
Example  2.  Find  the  limit  as  a?  £=  00  of 


f.x)  = 


^x  +  \' 


which  may  be  said  to  lead  to  the  symbol  00  /oo . 

Dividing  numerator  and  denominator  by  x^,  we  see  that 


limit  fix)  ^   limit        4  -  d/x  4-  l/x'' 


a;  zz  00  X 


:b  00      1  _  5/aj  _^  i/x' 


^=4. 


io8  LIMITS;   INDETERMINATE  FORMS,  [Ch.  XII 

Example  3.  The  limit,  as  x  approaches  3,  of 

^3^-  3   _  v/3  (  V^-  l/3") 


«-3  (  |/a.+  |/3)(  \/x-  |/3) 

is  |/3"/2  1^3  H  i     For  a;  =  3,  the  fraction  leads  to  the  symbol  0/0. 

EXERCISES. 

State  which  indeterminate  form  arises  for  the  value  of  x  given  in  each  of 
the  following  problems.  Find  the  limit  of  the  function  as  x  approaches  the 
value  given. 

.     Vx-1. 


for  x  =  l.  6.  ^ "—^  for  x 


V^-^  '  ^  -  ^        2  -  V^x 

„    (x  —  d){7x  +  1)  .  x^  -h^x 

11.  g-^^^  +  l  for  ^  =  2.  12.    i^I+5^_^J^El  for  .^.  =  -  1. 

X  —  2  1-i-x 

103.  In  giviog  a  complete  definition  of  limit,  which  shall  give 
not  only  a  necessary  condition  for  a  limit,  but  the  necessary  and 
the  sufficient  conditions,  we  give  in  this  section  the  definition  of  a 
limit  as  7t  =  00  ,  and  in  §  104  the  definition  of  a  limit  as  :c  =  «,  where 
a  is  a  given  constant.  The  former  case  is  the  one  used  in  the  later 
chapters  of  the  text. 

In  giving  a  necessary  condition  for  a  Umit  (§  100),  nothing  was 
said  of  the  effect  arising  by  Jetting  x  approach  a  by  different 
methods,  for  example,  by  passing  through  positive  values  only, 
negative  values  only,  or  rational  values  only,  or  eveii  integral 
values  only,  etc.  Thus  the  sum  of  an  even  number  of  terms  of 
the  geometrical  progression  1,  —  1,  +  I^  —  1>  +  l^---  is  zero, 
the  sum  of  an  odd  number  of  terms  is  +  1-     As  the  number  of 


Sec.  103]  COLLEGE  ALGEBRA,  109 

terms  is  increased  without  bound,  there  is  no  limit  approached. 
If  n  grows  large  through  even  values  only,  the  necessary  condition 
of  §  100  for  the  limit  zero  is  satisfied,  whereas  we  do  not  admit  that 
there  exists  a  limit  of  the  progression. 
In  §  68  was  discussed  the  sum 

and  it  was  shown  that  the  difference  between  S^  and  2  may  be  made 
as  small  as  we  please  by  choosing  n  sufficiently  large.    Thus  to  make 

¥^^  ^  Too '  ^*  ^"^^^^  ^^  *^^^  n>7]  to  make  — --  <  j^,  it  suf- 
fices  to  take  n  >  10.  Then  Sn  is  said  to  have  the  limit  2  as  n  in- 
creases indefinitely. 

Definition.*  If  the  function /(w)  differs  from  a  constant  I  by  an 
amount  less  than  an  assigned  positive  number  e,  however  small, 
for  all  values  of  n  which  exceed  an  assignable  number  (whose  value 
depends  upon  the  value  of  e),  then/(?^)  is  said  to  have  the  limit  I  as 
n  increases  indefinitely  and  we  write 

limit    f{?i)  =  L 

71  ±    CO 

If,  in  the  preceding  example,  e  be  y^^,  then  2  —  Sn  <  e  for 
n  >  7.     If  e  be  yoVo'  then  2  —  Sn  <  e  torn  >  10. 

Consider  the  repeating  decimal  .3  =  .333  .  .  .  and  set 
Sn^  .3333  ...  (to  n  decimal  places). 
For  er^.OOl,  ^-S.-^^-,^  <  e.     For  e^.OOOOl,  ^-6;=  ^^oVtto <^. 

In  general,  for  e  =  10~%  |  —  aS'^  <  e  f or  every  71  ^  e.     Hence 

limit  Sn  =  i- 

71  iz   CO 

*For  an  infinite  limit  I  =  od  ,  the  definition  would  read  :  If/(«)  exceeds 
an  assigned  positive  number  E,  however  great,  for  all  values  of  n  which 
exceed  an  assignable  number,  then 

limit     f{n)  =  00 . 
n  ^  00 


no  LIMITS;    INDETERMINATE  FORMS.  [Ch.  XII 

The  limit  of  .6  e  .666  ...  is  |,  being  twice  the  limit  of  .3,  a  result 
following  from  the  theorem  : 

T/',  as  71  increases  indefinitely^  f{n)  lias  the  limit  I,  and  if  cis  amj 
constant,  then  c  » f{n)  has  the  limit  c  .  I. 

In  fact,  for  an  assigned  positive  number  e',  however  small, /(/z) 
differs  from  I  by  an  amount  <  e'  for  every  value  of  n  whicli  exceeds 
an    assignable   number.       Hence   c .  f\n)  differs   from   c  ,1  by  an 

amount  <  ce'  for  those  values  of  7i.     Taking  e'  =  — ,  we  conclude 

that,  for  an  assigned  positive  number  e,  c  .f{n)  differs  from  c  .  Z  by 
an  amount  <  e  for  every  value  of  n  which  exceeds  aa  assignable 
nuQiber,  so  that  c  ,f{n)  has  the  limit  c .  I. 

104.  By  way  of  introduction  to  the  general  definition  below,  a 
function  having  a  limit  and  a  function  not  having  a  limit  as  x 
approaches  3  are  first  given. 

The   limit  ot  2x  -\-  4:  as  x  approaches  3  is   10.      In  fact,   10 

and   2x  +  4  differ  by  an  amount  <  e  for  every  x  between  3  —  — 

and3  +  -|. 

Let  a  function  E{x)  be  defined  for  positive  values  as  follows  : 
E{x)  =  X,  for  X  an  integer;  B{x)  equals  the  integer  just  greater 
than  X,  foYX  fractional  or  irrational.  Hence  as  x  increases  from  2.1 
up  to  3,  including  3,  £J{x)  remains  equal  to  3;  but  as  x  decreases 
from  4  toward  3,  excluding  3,  B(x)  remains  equal  to  4.  Thus  E{x) 
does  not  have  a  limit  as  x  approaches  3.* 

General  definition  of  limit.  If  the  function /(.t)  differs  from  a 
constant  I  by  an  amount  less  than  an  assigned  positive  number  e, 

*In  a  more  restricted  sense  of  limit,  as  x  approaches  3  through  vahies 
immediately   less  than   d,.E{x)  has  the  limit  3.     Similarly,  by   §101,  as  x 

approaches  0  through  positive  values,  -   has  the  limit  +  oo  ;  as  ^c  approaches 

10 
0  through  negative  values,  —    has  tlie  limit  —  oo. 


3ec.  105]  COLLEGE  ALGEBRA.  IM 

however  small,  for  all  values  of  x  between  a  —  6  and  a-\-  6  (except, 
perhaps,  x  —  a),  where  c^  is  a  constant  and  d  an  assignable  positive 
number  depending  upon  e,  then  f(x)  is  said  to  have  the  limit  Z  as  a; 
approach es  a,  and  we  write 

limit  f{x)  =  I, 
x  —  a 

For  example,  tlie  limit  of  the  above  function  2.^•  +  4  as  a;  ap- 
proaches 0  is  4,  since  the  difference  2x  is  numerically  <  e  for  every 

r  between  —  —  and  -f  i^- 

The  definition  is  to  be  modified  if  either  a  or  /  is  infinite  or  if 
both  are  infinite.  Thus,  if/(:^")  exceeds  an  assigned  positive  number 
E,  however  great,  for  all  values  of  x  between  a  —  d  and  a-\-6, 
where  «5  is  a  constant  and  d  is  an  assignable  positive  number,  then 
f{x)  has  the  limit  -j-  go  as  :?;  approaches  a,  and  we  write 

limit    f{x)  =  4"  ^  - 
X  i=  a 

Similarly,  the  limit  —  co  may  occur.  The  case  «  =  oo  is  treated 
in  §  103.      For  example, 

limit    1  _  limit     4x  —  3  _    limit      4  —  3/x  _ 

X  i=:  0  '^2-'^^      n-  =.  CO     2x+  1  ~  ^  =  00     2  -f  1/x  ~    * 

105.  For  all  values  of  x  different  from  a,  the  functions 

f{^)=  ^.^,Zl^     F{x)^x+a 

are  equal.  But /(a:)  is  not  defined  for  x  =  a,  since  it  then  iassumes 
the  indeterminate  form  ^.  B\\tf{x)  differs  from  2a  by  less  than  e 
for  all  values  of  x  between  a  —  e  and  a  -\-  c.     Hence 

limit /(2:)  =  2«  —  limit  F{x). 
X  ±  a  X  ±  a 

In  a  similar  manner,  we  may  prove  the  general  theorem : 
If  tioo  functions  of  x  are  always  equal  ivhenever  each  is  defined 
and  if  each  function  has  a  limit  as  x^a^  the  tivo  limits  are  equal. 


112  LIMITS;    INDETERMINATE  FORMS.  [Ch.  XII 

Suppose  that  two  fLinctions  f{x)  and  ^ (re)  have  the  respective 
limits  F  and  G  'd^  x  approaches  a.  Then  for  an  assigned  positive 
number  e/2,  however  small,  the  difference  between /(:c)  and  t  anc 
the  difference  between  g{x)  and  G  are  ea^h  <  e/2  for  all  values  of  iij 
between  a  —  S  and  a-\-  6,  where  d  is  an  assignable  positive  number,^ 
Hence  the  difference  between /(a:)  -\-  g{x)  and  i^+  6^  is  <  e  for  al 
values  of  x  between  a  —  6  and  a  -\-  d,  which  proves  the  first  of  th( 
following  theorems: 

The  limit  of  the  su7)i  of  two  functio7is  equals  the  sum  of  theii 
limits.    Ihe  limit  of  the  difference  equals  the  diff'erence  of  the  limits. 

Denote  by  |  7i  |  the  numerical  value  of  7i,  whether  n  is  positive  oij 
negative.     We  may  write 

I  f{x)g{x)  -  FG\^\  f{x)g{x)  -f{x)G\^\  f{x)  G-FG\ 
^mcQ.  f[x)  has  the  limit  F,  and  g{x)  the  limit  G,  then 


I 


1/(^)1-1^1  <e,  \f{x)-F\<^^^,  \g{x)-G\<^-_^ 

for  all  values  of  x  between  a  —  S  and  a-{-  6,  where  d  is  a  suitably 
chosen  positive  number.     Hence 

\fix)g{x)-FG\^^^^,^^^^<~  +  ^. 

Since  the  difference  of  f(x)g(x)  and  FG  is  less  than  e  for  all  values 
of  x  between  a  —  d  and  a-{-  6, 

limit  /(:?:)^(^)  ==  FG  =  limit /(:r)  .  limit  g{x). 
X  ^  a  X  ^  a  X  ±  a 

We  have  therefore  established  the  following  theorem  : 

TJie  limit  as  x±a  of  the  product  of  tiuo  functions  of  x^  each  oj| 
which  has  a  limit  as  Xzta.,  equals  the  product  of  their  limits. 

Similarly,  we  may  prove  the  theorem : 

The  limit  of  the  quotient  of  tivo  fu7ictions  equals  the  quotient 
of  their  limits,  if  the  limit  of  the  divisor  is  7iot  zero. 


CHAPTER   XIIL 

CONVERGENCY   AND   DIVERGENCY   OF  SERIES. 

106.  It  was  shown  in  Chapter  VI  that  any  term  of  an  arith- 
\  metical  progression  may  be  derived  from  the  first  term  and  the  com- 
I  mon  difference,  also  that  any  term  of  a  geometrical  progression  may 
I  be  derived  from  the  first  term  and  the  common  ratio.  These  progres- 
i  sions  are  examples  of  series  in  that  they  give  a  succession  of  quan- 
tities formed  according  to  some  definite  law.  In  Chapters  VIII  and 
XI  were  considered  other  series,  as  1-24-^-3  ~{-  3-4:  -\-  ,  .. 

If  the  number  of  terms  of  a  series  be  finite,  it  is  called  a  finite 
series  ;  if  the  series  does  not  terminate,  it  is  called  an  infinite  series. 
To  form  the  sum  of  a  finite  series  is  theoretically  an  elementary 
problem  and  the  order  in  which  the  terms  are  added  is  immaterial. 
In  the  case  of  an  infinite  series,  it  is  necessary  to  define  what  is  to  be 
understood  by  the  expression  '^sum  of  an  infinite  series."  We 
consider  the  sum  S^  of  the  first  n  terms  of  the  series  and  let  n 
increase  without  bound.  For  the  series  1^ -f  ^^  +  3^  +  .  .  .  ,  the 
sum  Sn  increases  without  bound  when  n  does.  For  the  series 
l-|-i+l+8  +  -'-^  ^^^^  s^^^  ^n  has  the  limit  2  when  7i  increases 
without  bound  {§  103).  For  the  series  l~l-f-l— 1  +  ...,  the 
sum  Sn  is  1  or  0  according  as  w  is  odd  or  even,  so  that  S^  does  not 
approach  a  limit  as  n  increases  without  bound. 

Definitions.*     An  infinite  series  is  called  convergent  if  the  sum 

*  In  problems  involving  infinite  series,  it  is  necessary  to  know  whether  the 
series  is  convergent  or  divergent.  Thus,  if  S  denote  the  sum  of  a  given  series, 
we  can  conclude  from  a  -\-  S  =  b  -\-  S  that  a  =  b  if  the  series  be  convergent, 
but  7iot  if  it  be  divergent. 

I 


114  CONVERGENCY  AND  DIVERGENCY  OF  SERIES.   [Cn.  XII 

Sn  of  its  first  71  terms  approaches  a  finite  limit  as  n  increases  witliou 
bound,  divergent  if  S^  does  not  approach  a  finite  limit. 

Of  the  preceding  examples  of  infinite  series,  1  +  i  +  i  +  •  .  •  i^ 
convergent,  while  1^  +  2^  +  3'^  +  .  .  .  and  1  -  1  +  1  -  1  -f  .  .  . 
are  divergent. 

By  definition,  a  series   is  convergent  if,  and   only   if,  limit  S, 

7^  =  GO 

exists  and  is  a  finite  number  S.  By  §  103,  the  condition  is  thai 
the  difference  between  aS^„  and  S  shall  be  less  than  an  assigned  posi- 
tive number  e,  however  small,  for  all  integers  n  greater  than  ar 
assignable  positive  integer.     Hence  must* 

I  /S'  -  /^n  I    <   6 

for  every  integer  n  greater  than  an  assignable  positive  integer. 

107.   Consider  the  infinite  geometrical  progression  1,  x,  x'^,  ,  .  . 
Then 

1  x"" 


Sr,=  l+    X+X^  +  .    .    .    +  X^-'  = 


1  —  X         I    —  X 


If  |a;|  >  1,  the  absolute  value  of  x""  increases  without  bound  as  n 
does,  so  that  the  series  is  divergent.  If  1 2:  |  <  1,  x""  has  the  limit  C 
as  n  increases  without  bound  (§  69).     We  may  show  tliat 

limit  a   _      ^ 


n^  00  1  —  x 

To  take  a  numerical  case,  let  a:  =  |,  so  that  8n-=  ^  —  3(f)^     To 
make  3  —  S^  <  e,  it  is  necessary  and  sufficient  to  take 

log  3  —  log  € 
^  ^  log  3  -  log  2* 

There  remain  the  cases  x  =  ±  1.     For  x  =  -{-  1,  S,^  =  n  and 

limit  Sn  =  limit  n  =  co  , 
n  :^  CO  n  =  00 

*  As  explained  before,  |  t  I  denotes  the  mimencal  value  of  the  real  number 
t ;  thus  1  +  2  I  =  2,  I  —  2  I  =^  2.     The  symbol  U  |  is  read  absolute  value  of  t. 


Sec.  108]  COLLEGE  ALGEBRA.  115 

For  a;  =  —  1,  Sn  is  1  or  0  according  as  n  is  odd  or  even,  and  hence 

the  series  1  —  l  +  l—  l+---is  divergent.     Note  that is 

now  \.     We  have  therefore  proved  the  theorem : 

The  infinite  series  \  -\-  x -\-  x^  -\-  .  .  .is  cojivergent  if\x\  <1,  and 

is  divergent  if  \x\  ~  1. 

An  infinite  arithmetical  progression  is  always  a  divergent  series. 
[f  a  be  the  first  term  and  d  the  common  difference,  the  ni\\  term 
^„  is  «  +  {n  —  l)d  and  the  snm  of  the  first  n  terms  is  (§  63) 

'Sn  =  ^n{a  +  tn)  =  in[2a  +  {n  -l)d]. 

.',     limit  Syt  =^  00  . 
n  =  CO 
Hence  the  series  is  divergent.     Since  limit  ^^  =  qo  ,  the  preceding 

result  may  also  be  derived  from  the  tlieorem : 

In  a  convergent  series,  the  limit  of  the  nth  term,  as  n  increases 
wit  ho  21 1  hound,  is  zero. 

In  proof,,  we  note  that  S^  and  S^^^  both  liave  the  limit  S^  so 
.hat  their  difference  4  l^^s  the.limit  zero  (§  105). 

108.    Comparison  Test.     Of  ttvo  series  witli  all  terms  'positive, 

^'■i  +  «2    +  «3  +  •    •    •   ^  ^'Z  +  «2'  +  ^'Z  +   •    •    •   :» 

\et  the  first  be  convergent.     If  a\^  ~,  a,^  ^  from  an  assigned  value  of 

n,  omvards^  then  the  second  series  is  coywergent. 
Let  m  be  the  assigned  value  of  n,  so  that 

^4  <  ^«  (for  any  n  ~Z  m), 

[f  S\  denote  the  sum  of  the  first  n  terms  of  a-^'  -\- ci^'  -{-  •  •  •  > 
ind  S,,  the  sum  of  the  first  n  terms  of  a^-[-  a ^ -{-,..  ,  with  the 
.imit  8,  then 

^'n  ~  ^m    ^     Sn  —    Sm  <  S  —  S^* 


Il6  CONVERGENCY  AND  DIVERGENCY  OF  SERIES,   [Ch.  XIH 

Hence  S^  remains  finite  for  every  value  of  n.  But  8'  is  a  posi- 
tive variable  which  increases  as  n  increases.  Hence  there  exists 
number,  say  8',  toward  which  8'  approaches  arbitrarily  near  bu 
does  not  pass.     Thus  8'^  has  the  limit  ;S''. 

Comparison  Test.      Of  two  series  loitli  all  terms  positive,  | 

^1  +  ^2  +  ^3  +  •  •  •  >         ^i'  +  ^2'  +  «/  +  •  •  •  ^ 

let  the  first  he  divergent.     If  a^  y  an,  from  an  assigned  value  of 

n  omvards,  the  second  series  is  divergent. 
With  the  preceding  notations,  we  have 

'     '    K"  ^^m  ^  ^n  -  ^m  (any  n  =  m).\ 

Since  8^  is  always  positive  but  does  not  approach  a  finite  limit,  it 
must  increase  without  bound  as  n  does.     Hence 

limit  >S^^  Tz  limit  8^  -\-  8!^  —  8^  ~  ^  - 
n-^co  n-:^  (JO 

109.  The  simplest  harmonical  progression  is  1,  ^,  ^,  ^,  , 
Since  the  wth  term  -  approaches  zero  as  71  increases  without  bound, 
so  that  the  individual  terms  are  ultimately  very  small,  it  might  be 
supposed  that  the  series  would  have  a  finite  sum  and  hence  be  con- 
vergent.    However,  by  comparing  the  two  series 

l+i+l  +  i  +  i  +  i  +  i  +  l+TV-f.--    +tV  +  .  . 

i  +  i  +  i  +  }  +  i  +  f  +  l  +  i+ !+•••  + iV  +  .- 

we  observe  that  each  term  of  the  second  is  equal  to  or  greater  than 
the  corresponding  term  of  the  first.  If  the  first  is  divergent,  the 
second  is  divergent  (§  108).     But  the  first  equals 

and  is*  divergent.  To  show  that  there  is  a  boundless  number  of 
terms  ^  in  the  last  series,  we  observe  that  a  term  |  was  derived  fromj 
each  of  the  parenthesized  groups  of 

1  +  (i)  +  a+ i)  +  a+ i  +  ^  + 1)  +  (i+ •  •  •  +  iV)  +  •  •  • » 


3ec.  110]  COLLEGE  ALGEBRA,  II 7 

1111 
he  final  terms  of  the  groups  being  ■^,  k^^  ^^  ^y  '  •  '  y  so  ^^^.t the 

lumber  of  groups  is  boundless.     Hence  the  series 

s  a  divergent  series, 

110.   As  a  generalization  of  the  preceding  theorem,  the  series 

2)  1+1+1  +  1+  +1  + 

^/  ^p  ~^  2^       d^       4:^  n^ 

s  divergent  if  p  'Z  1  and  convergent  if  p  >  1. 

Let  first  ^  <  1  (including  the  case/>  negative).     Then 

—  >  —       (for  any  positive  integer  n>l), 

'ndeed,  if  both  terms  of  the  inequality  be  multiplied  by  n^  it 
)ecomes  n^~^  >  1  and  hence  is  true,  since  1  —  p  is  positive.  Com- 
)aring  the  two  series  (1)  and  (2),  we  have  shown  that  each  term  of 
2),  after  the  first,  is  greater  than  the  corresponding  term  of  (1), 
vhich  is  divergent.  By  §  108,  series  (2)  is  divergent. 
Let  next  jt?  >  1.     Comparing  (2)  with  the  series 

r)        1+1  +  1  +  1+1+1  +  1  +  1  +  ..., 

2^       2^       4^       4^       4^       4^       8^ 
lach  term  of  (2)  is  equal  to  or  less  than  tlie  corresponding  term  of 

2'),  thus  3^  <  ^.     5?  <  4?'  •  •  •     I^^^^®  ^^  (^')  ^^  convergent, 
2)  will  be  convergent  (§  108).     But  (2')  may  be  written 

1  +  1+1  +  1+     -i^fi_l) 

2 
>eing  an  infinite  geometrical  series  with  the  common  ratio  — ,  which 

3  less  than  unity  since  jt?  >  1. 


li8  COM/ERGENCr  AND  DiyERGENCY  OF  SERIES.   [Ch.  XII: 

111.  The  series  (2),  including  (1)  as  a  special  case,  are  stand 
ard  series,  by  a  comparison  with  which  a  given  series  may  ol'tei 
be  proved  convergent  or  divergent,  as  the  case  may  be. 
Example  1.  Each  term  of  the  series 

1  ^  4  ^  9  ^  10  ^  ^      TiV  ^ 

is  greater  than  the  corresponding  term  of  series  (1),  since 
71  +  1        1     ,     1         1 
n^  n       n^       n 

Since  (1)  is  divergent,  the  given  series  is  divergent. 
Example  2.  The  series  whose  ;/th  term  is 

_  w  +  3 
n^  +  1 

,        ^       f      ,  71  +  3   _  7i  +  3/2  4  „ 

IS  convergent,     m  tact,  Un  <  — —t. — ., —  —  — .     Hence   each  term  c 

7L'^      <       n-^       <  n^ 

the  given  series  is  less  than  the  corresponding  term  of 

4        4  4 

j^2  +  2"2  "^~  •  •  •  "^  ;^  +  •  •  • ' 

which  is  convergent,  being  4  times  series  (2)  for  j9  =  2. 
Example  3.  For  positive  values  of  x  and  a,  the  series 

X       X  -}-  a      X  -\-  2a  x  -\-  7ia 

X 

is  divergent.     Multiplying  2£  by  a  and  setting  y  =  -,  we  get 

Let  the  integer  just  greater  than  y  be  m.     Then 

a:S       -4-       ^  1        ,  ,         1 

m      m  -\-  i      m  +  2  m  -\-  m 

the  latter  being  only  a  part  of  the  divergent  series  (1). 

EXERCISES. 

Test  the  series  for  convergency  or  divergency: 

1    -  -I-  -  +  ...  -I = f-  .  .  .       2.     -  -4-  —  4-  .  .  .  +  -i-  -f-  .  .  . 


1.2'^2.3"^  ^//(n-fl)"^*"       '  1.2.3~^2-3  4'^""^n(/i-f-l)(7i-f 2)  ' 


Sec.  112]  COLLEGE  ALGEBRA,  1 19 

fi   -La- J-4--l-,_^  e      ^     I      ^     I      ^     I 

3-42  "^4.5*''^  5-62 '^*  *  *  3'-'-42'^  4^-52'^  5^-6"^  "^  **  * 

1^3^  5^  •lP^2P^3i>^  ^     nP     ^ 

9. 1 7—r. 7T  +  / — T-TTT — i— k;  -[-•••  for  a;  and  y  positive. 

10.   1  +  2  -  3  +  1  +  2  -  3  +  .  .  . 

112.  An  infiyiite  series  is  convergent  if  the  terms  are  alternately 
positive  and  negative,  if  each  term  is  numerically  less  than  the  pre- 
ceding, and  if  the  nth  term  has  the  limit  zero  as  n±  cc, 
1       In  the  series  u^  —  u^  +  21^  —  ii^  +  .  .  . ,  let  each  tc  be  positive 
and  let  * 

^        ^        .         .  limit  ri 

2l,>  u,>  zf,>  u^>  ,.,  ,  ^^^u^  =  0. 

By  considering  the  sum  of  an  even  number  of  terms, 

I  S,n  =  {^h  -  ^2)  +  (^s  -  '^4)  +  •  •  •  +  (^2n  ^1  -  W2J, 

we  observe  that  ASgn  is  positive.     Hence 

^2n  +  1  =  ^2n  "T   ^2»  +  1 

is  also  positive.     Writing  S2n  + 1  in  the  form 

we  see  that  Szn  +  i  <  w^.  Hence  /S^2n  <  '^^-  Since  the  quantities  S^, 
S^,  .  ,  .  82^,  '  '  '  ^1*6  increasing  positive  numbers,  each  <  n^,  Sc^ 
must  have  a  finite  limit  I  as  n  increases  without  bound.     But 

limit   (a  q  \  —  li^^i^   ./  —  o 

Hence  /S^2n  +  i  ^^so  approaches  the  limit  /.  The  series  ti^  —  1/2  +  •  •  • 
is  therefore  convergent. 

Example.  The  series  1— i  +  i  —  i  +  .-.  is  convergent.    But 

12^3       4^ 
*  The  student  should  make  the  proof  for  the  example  1— ^  +  i  —  i  +  ..» 


I20  COhlVERGENCY  AND  DIVERGENCY  OF  SERIES.  [Ch.  XIII 


is  divergent  (see  end  of  §  107),  since 

limit   n-\-l 


=  1. 


For  the  latter  series  the  third  condition  of  the  theorem  is  not  satisfied,  while 
the  first  and  second  conditions  are  satisfied. 

113.  In  comparing  a  given  series  with  a  standard  series,  it  is 
often  convenient  to  remove  a  Ji?iite  number  of  terms  of  the  given 
series,  the  removal  of  which  subtracts  only  a  finite  quantity  and 
hence  does  not  alter  the  convergency  or  the  divergency  of  the  given 
series.     Thus  in 

it  is  convenient  to  remove  tlie  first  five  terms,  whose  sum  is  finite. 
There  remains  the  infinite  series 


5!i^  +  6 ' 

5'r,  .5 


<5! 


[^  +  ^©^(I)V-} 


The  series  in  brackets  is  an  infinite  geometrical  progression  whose 
sum  is g  =  6.     Hence  the  series  (3)  is  convergent  (§  108). 

114.  Theorem.  An  infinite  series  all  of  loliose  terms  are  positive 
is  convergent  if,  after  any  particular  tenn,  the  ratio  of  each  term 
to  the  preceding  is  alioays  less  than  some  fixed  quantity  lohich  is 
itself  less  than  unity. 

The  terms  preceding  the  particular  term  in  question  may  be 
removed  without  altering  the  convergency  of  the  series,  since  their 
number  is  supposed  to  be  finite.  Let  the  remaining  series  be  de- 
noted by 

u^  +  n^  +  Uz  +  tt,-T  .  .  ., 

and  let  -^  <  r,     -'  <  r,     -*  <  n  ^  .  ~     f r  <  1). 

u,  u,  u^ 


Sec.  115]  COLLEGE  ALGEBRA.  121 

Multiplying  together  the  first  two  inequalities,  the  first  three,  etc., 
we  find  that  —  <  r^,     — *  <  r^,  etc.     Hence 

All  the  quantities  being  positive,  we  find  by  addition  that 

^^  +  W'2  H-  ^8  +  «^4  +  •  •  •  <  ^1(1  +  r  +  r^  +  r^.+  .  ,  .), 
The  infinite  geometrical  progression  in  parenthesis  has  the  sum 

,  since   r  <  1.      Hence    v^  -\-  u^ -{-..»  -{-  itn  is  a   positive 

1  —  r 

quantity  which  increases  as  11  increases,  but  remains  <      2_  ' ->  ^^^ 

hence  has  a  finite  limit. 

But  nothing  follows  in  the  case  r  =  1,  since  1+^'+^^+  •  •  •  H-r*"""' 
increases  without  bound  as  n  does.  Although  u^-\-  u^-\- ,  ,  . -\-  ti^ 
is  less  than  the  former  sum,  it  may  have  a  finite  limit  or  increase 
without  bound. 

115.  Theorem.  A  series  luitli  positive  and  negative  terms  is 
convergent  if  the  series  derived  from  it  ly  making  all  the  terms 
positive  is  convergent. 

Since  the  series  with  all  terms  positive  has  a  finite  sum  ^,  the 
sum  Sr,  of  the  first  n  terms  of  the  given  series  lies  between  —  ^ 
and  +  2.  Hence,  if  we  show  that  S,^  approaches  a  limit  as  n 
increases  indefinitely,  this  limit  will  be  finite  and  the  series  conver- 
gent. Let  Pp  be  the  sum  of  the  p  positive  terms  in  8n,  and  Nq  the 
sum  of  the  q  negative  terms  in  Sn  after  their  signs  are  made  posi- 
tive.    Then 

S,  =  Pp-Nq         {p  +  q  =  n). 

Also  Pp  and  N^  are  each  positive  and  less  than  ^.  Hence  Pp  ap- 
proaches a  fixed  value  P,  and  N^  a  fixed  value  A^  as  ^  increases 
without  bound,  so  that  either  p  or  q  increases  or  both  increase 
without  bound.     Hence 

limit  Sn  =  P  —  N—  fixed  finite  number. 

W£r  CO 


122  CONVERGENCY  AND  DIVERGENCY  OF  SERIES,   [Ch.  XIII 

116.  In  view  of  the  preceding  theorem,  the  result  of  §  114  leads 
to  the  more  general  theorem : 

Eatio  Test.  A  series  is  convergent  if,  after  any  particular  term^ 
the  absolute  value  of  the  ratio  of  each  term  to  the  'preceding  is  always 
less  than  some  fixed  quantity  ivhich  is  less  than  unity. 

Thus,  in  the  series  (3)  of  §  113,  the  7ith  and  n  +  1st  terms  are 

5n-l  5^ 


{n-l)V      n\ 

The  ratio  of  the  latter  to  the  former  is  5/w.     Hence,  if  7^  >  5,  the 

ratio  is  at  most  |  and  hence  is  always  <  \\,  for  example.     Hence 

the  series  is  convergent.    Note  that  here  also  5  terms  were  removed. 

For  the  series  (1),  the  ratio  of  the  ni\i  term  to  the  preceding 

term  is  — \ =  =  1 .    While  this  ratio  is  less  than 

71      n  —  1  n  n 

unity  for  every  value  of  n,  it  is  impossible  to  find  a  proper  fraction 

/"such  that  1 <  fior  every  n.     Indeed,  as^  =  Go,  1 il. 

To  put  the  matter  in  another  light,  if  one  assigns  to/ the  value 

.9999,  the  ratio  1 will  exceed  /  =  .9999  as  soon  as  n  >  10000. 

'  n  ^ 

Hence  the  theorem  fails  to  prove  the  convergency  of  1  -)-  i  +  i  +  • . . 
By  §  109,  the  series  is  divergent. 

Series  (3)  is  a  special  case  of  the  series 

The  ratio  of  the  {n  +  l)st  term  to  the  nth.  term  is  x/71  and  is,  in 
absolute  value,  less  than  a  quantity  less  than  unity  for  all  values  of 
n  greater  than  x.  Hence,  if  X  be  the  least  positive  integer  equal 
to  or  greater  than  |  a:  | ,  we  need  only  remove  the  first  JT  terms  of  the 
series,  so  that  in  the  remaining  series  the  ratio  of  each  term  to  the 


Sec.  117]  COLLEGE  ALGEBRA.  123 

preceding  shall  be  in  absolute  value  always  less  than  a  fixed  quan- 
tity less  than  unity.     Hence  series  (4)  is  convergent  for  every  x. 

117.  An  infinite  series  all  of  ivliose  terms  are  of  like  sign  is 
divergent  if,  after  any  particular  term,  the  ratio  of  each  term  to  the 
preceding  term  is  alivays  greater  than  or  equal  to  unity. 

Remove  the  terms  preceding  the  particular  term  in  question  and 
denote  the  remaining  infinite  series  by 

«^i  +  ^^2  +  ^^3+  •  •  • .  ^2  ^^^^  ^^3  ^  «^2^    •  •  • 
. •.    i^i  +  t<2  +  t^3  +  •  •  •  +  ^'n  ^  ^^  +  ^1  +  '^1  +  •  •  •  ^  nu^. 

Since  nu^  =  cc  as  /i  =  00,   u^  +  ^2  +  ^^3  +  •  •  •  is  divergent. 

For  example,  the  ratio  of  the  nth  term  to  the  {n  —  l)st  term  in 

172  "I"  2^3"  +  **  •■^(^-  \)n'^  n{n^l)  "*"  *  *  * 

\^  Z{n  —  1)  -T-  {n  -\-  1)  and  hence  is  greater  than  1  for  all  values  of 
n  greater  than  2.     Hence  the  series  is  divergent. 

Ratio  Test.  An  infi^iite  series  of^^ositive  and  negative  terms  is 
divergent  if,  after  any  particular  term,  the  ratio  of  each  term  to  the 
preceding  term  is  numerically  equal  to  or  greater  than  a  fixed  quan- 
tity r  >  1. 

Beginning  with  the  term  in  question,  let  the  series  be 
^1  +  ^a  +  ^'3  +  •  •  •  >  where,  for  every  n^ 

U„        > 


Hence  |  '^^n  + 1 1  c  ^"^  t  '^^i  I '  ^^  ^^^^  limit  1 1^^  ^  ^  |  =  00  .    But  a  series  is 
^  n±co 

divergent  unless  the  nth.  term  approaches  zero  as7uncreases  (§  107). 

118.  A  more  convenient  form  of  the  ratio  tests  of  §§  116, 117  is 

given  in  the  equivalent  theorem: 


124  CONyERGENCY  AND  DIVERGENCY  OF  SERIES,   [Ch.  XIII 

ffin  any  series  ii^  +  ?^2  +  ^s  +  •  •  •  there  exists  *  a  limit y 
__  limit 

the  series  is  convergent  if  I  <  1,  divergent  if  I  >  1.     Ifl  =  lya 
further  test  is  necessary. 

As  an  example,  consider  the  logarithmic  series  (Ch.  XVI), 

/yt  /yi  /yiO  /y«4  /ytfl  —  1  /y»H 

1        2  +  3     ,  4  +  •  •  •       ^      ^^        n-1       ^      ^^    n       '" 


The  ratio  of  the  ^^th  term  to  the  {71  —  l)st  term  is 

n  \        nJ 

Hence  limit  |  r  |  =  | ^ | ,  so  that  the  series  is  convergent  ii\x\  <  1 , 
but  divergent  if  |a;|  >  1.  For  x  =  -\-  1,  the  series  is  converg- 
ent by  §  112.  For  x  =  ^  1^  the  series  is  the  divergent  series 
--1-i-i-...  (see  §109). 

EXERCISES. 

1. —r  +  —7-^ ::: — h  .  •  •  IS  converffent  if  a  and  b  are  positive. 

a      a-{-b      a-{-2b      a-f-36 

»•  r2  -  2^  +  3^4  -  A  +  •  •  ■  '^  convergent. 

For  what  values  of  x  are  the  following  series  convergent  ? 

••|+i+?  +  ---+^  +  --      6.1  +  3.  +  3a.  +  4x3+... 

'•2+8+4+---  +  f+3  +  ---      *•  ^+4+---+-^  +  --- 

9.  *+3+5  +  y  +  .--  10.  l  +  3+3,  +  45  +  --- 

/ys       j>A      3.7  2*       3^      4* 

"•*-|]  +  il-f!+---  ^2.1  +  ^,  +  3-  +  j-,  +  ... 

*  This  limit  does  not  exist  for  the  series  1  -{- 2x-{-x'^ -\-2a^ -^x*  +  2x^  +  . .. 


CHAPTER  XIV. 

POWER   SERIES;    EXPANSIONS   INTO   SERIES. 

119.  An  infinite  series  of  the  form 
(1)  a^  +  a^x  +  a^x^  +  .  .  .  +  an-ix""-^  +«»^"  +  •  •  .  5 

in  which  each  a  is  independent  of  x,  is  called  a  power  series  in  x^ 
since  each  term  involves  x  merely  in  the  form  of  a  power.  It 
may  be  employed  in  an  investigation  only  when  it  is  convergent. 
To  apply  the  ratio  test  of  §118,  we  investigate  the  limit  of  the 

xct 
ratio  -^ — ^  as  n  increases  without  bound.     If  the  limit  exists,  its 

absolute  value  equals  \x\  -r-  I,  where 

limit 


71  —  00 

Noting  when  \x\  -r-  Z  is  less  than  1  or  greater  than  1,  we  find  that 
The  series  (1)  is  convergent  if  \'^\  <  I,  divergent  if  \^\  >  L 
For  the  case  \x\   =  /,  a  special  investigation  is  necessary. 
Corollary.  If  a  power  series  converges  for  x  =  a,  it  converges 

for  every  value  of  x  numerically  less  than  a, 

120.  Suppose  that  series  (1)  converges  iov  x  =  a^  a  4^  0.    Then 

iox  X  —  OL  the  first  of  the  following  limits  exists  and  is  finite: 

limit    /        ,        9  ,  ,        n\  limit    ,      .  ,  ,        „_|x 

^  3^  00     iV+^2^^+  •  •  •  -h«^n^")  =  ^  ^  ^  00     («l+  V  +  •  •  •  +  ^n^""      ')> 

the  equality  being  true  by  §  105.  Hence  the  series  a^  -\-  a^x  +  .  .  . 
is  convergent  forx  =  a  and  therefore,  by  the  preceding  corollary, 
for  every  x  numerically  less  than  a.     In  particular,  it  has  a  finite 

125 


126  POIVER  SERIES;    EXPANSIONS  INTO  SERIES.      [Ch.  XIV 

value  for  a;  =  0,  so  that  the  product  x{a^  -\-  a^x  -\-  ,  .  .)  vanishes 
for  a;  =  0.     We  may  therefore  state  the  theorem : 

If  the  series  a^  +  a^x  +  ci^x^  -\-  ,  .  .  is  convergent  for  a  value 
of  X  different  from  zero,  the  series  approaches  the  liniit  a^  as  x 
approaches  zero. 

121.  Theorem.     If  the  ttvo  infinite  series 

%  +  «l^  +  «2^^  +  •  •  •  +  "n^""  +  .  .  . , 
b,  +  \x  +  \x^  +  .  .  .  +  h^x^  +  .  .  . 

are  convergent  for  a;  =  o',  where  a  is  some  quantity  different  from 
zero,  and  if  the  series  are  equal  for  every  value  of  x  such  that 

r\^  «' L  then  a^  —  h^,  a^  =  b^,  ,  .  .  ,  a^  =  bn,  -  »  »  ,  and  the  ttvo 

series  are  identical. 

By  the  preceding  theorem,  the  series  approach  the  respective 
limits  a^  and  b^  as  x  approaches  0.     Hence  a^  =  b^.     Then 
a^x  +  «2^^  +  .  .  . ,     b^x  -{-  b^x^  +  .  .  . 

are  equal  and  are  convergent  for  x  ^a.     Hence 

a^  +  a^x  +  .  .  .,     ^^j^h.^x  +  .. 

are  equal  and  are  convergent  for  0  <    a:  p  a-  .     As  before,  these 

series  approach  the  respective  limits  «5j  and  Z>i  as  x  approaches  0. 
Hence  a^  =  b^     Proceeding  similarly,  a^  =  b^,  .  .  ..>  a^  =  b^,  .  .  . 

The  theorem  is  a  generalization  of  that  on  rational  integral 
functions,  §  78. 

Hence,  under  certain  conditions,  we  may  equate  the  coefficients 
of  like  powers  of  x  in  two  power  series.  This  principle  may  be 
applied  to  the  solution  of  problems  involving  infinite  series  with 
undetermined  coefficients. 

122.  The  expansion  of  a  function  into  a  power  series  is  valid 
only  when  the  resulting  series  is  convergent.     For  example, 

1 


l-x' 


\  '\-  x  -\-  x^  -\-  x^  -\-  ,  ,  .     ad  in^mfum, 


Sec.  132]  COLLEGE  ALGEBRA,  127 

are  equal  ^i  \x\   <  1,  but  not  if  |  ^  |  >  1.     Thus,   for  x  =  2,  the' 
fraction  equals  —  1,  while  tlie  series  has  an  infinite  sum. 

Example  1.    Expand  (1  —  ic)/(l  +  ^^)  ii^^o  a  power  series. 

We  seek  a  power  series  convergent  for  values. of  x  such  that  p  I  ^    a  , 

and  equal  to  the  given  fraction  for  those  values  of  x.     Employing  undeter- 
mined coefficients,  we  set 

1  —  X 

— -— 2  =  <^  +  ^^  -h  ^^''  +  d^^  -\-  ex*  -{-  .  .  , 

Multiplying  both  members  by  1  +  ^^,  we  have 

I  -  X  =  a  -{-  bx-i-  {a-i-  c)x^  +  (5  +  d)x^  +  (c  +  e)x*  +  .  •  • 
for  all  values  of  x  for  which  the  series  is  convergent.     Then 

a  =  1,  b  -  -  1,  a  +  c  =  0,  b  -\-  d  =  0,  c  -^  e  =  0,  .  .  . 


1  —   X 

.^  z:z  1  —  X  —  Q.^  -\-  x^  -^  X*  -}-  .  ,  .     (when  convergent). 


I  ...„....„..„,.,.. .„„._„„ 

the  more  direct  derivation  of  the  series  by  means  of  the  expansion 

I  -     ;-^  =  (i--)(r3^,-.))  =  (i--)a--^  +  ^^--«+---). 

holding  for  a;^  <  1,  namely,  for  —  1  <  x  <  -\- 1. 

Example  2.  Expand  \^1  —  x  into  a  power  series. 

Assume  4/I  —  :c  =  a  +  bx  -\-  cx^  +  dx^  -\-  ex*  -{- .  .  . 
For  values  of  x  for  which  the  series  is  convergent  after  its  terms  are  made 
positive,  the  square  of  the  series  may  be  found  by  the  elementary  rule  for 
squaring  multinomials.     Hence,  for  such  values  of  x, 

»1  -x  =  0"  +  2abx  -{-  (62  +  2ac)x'  +  {2ad  +  2bc)x^  4-  •  •  • 
.  •.  a"^  =  1,     2ab  =  -  1,     b"^  +  2ac  =  0,     2ad  -f  2^>c  =  0,  .  .  . 

For  the  positive  square  root,  a  —  ^  1.     Then  b  =  —  ^,  c=  —  I,  d  =  —  -^^^ 


I 


In  the  next  chapter  this  series  is  shown  to  be  convergent  if  j^l  <  1. 


5a;2  ^  4^^  _|_  3 
1  -\-  X  -\-  x^  -\-  x^ 
By  the  method  of  partial  fractions  (Chapter  VIII),  we  get 


Example  3.  Expand  ^ — r^;ri~:^\ — 3  ^^^^  ^  power  series. 


5^2  -f  4.^  -p  3  _      2  3.^  4- 1 

^  -  (1  +  x)(l  -{-x^)~  l-\-x~^  l  +  x^' 


T28  POIVER  SERIES;    EXPANSIONS  INTO  SERIES.       [Cn.  XH 

Then,  if  x^  <  1,  the  following  expansions  are  valid: 

^  =  2|1  ^  a^  +  a:^  -  a!3+ ...  +  (- l)na;n  + ..  .}, 

j^  =  (3a;  +  1){1 -cc^  +  a^ -...  +  (- 1)^3^2- 4- ... }. 

.•./=3  +  aj  +  ic2_5a,3_^...^j2_f_(_i)n|a.2n_|_|_2  +  3(_i)n}a,2n  +  i_^... 

123.  The  reversion  of  a  series  y  =  a^  +  a^x  +  a^x^  +  .  .  . 
consists  in  expressing  a;  as  a  series  in  ascending  powers  of  y, 
X  —  'b^-\-'h^y  ^  l^y^  +  •  •  •  "The  method  is  applied  only  to  conver- 
gent series  and  the  solution  is  valid  only  when  the  resulting  series 
is  convergent.  If  a^  =  0,  then  ?/  =  0  if  a;  =  0  (§120),  so  that 
h^  —  0.     If  a^  ^  ^^%Qi  y  —  a^  —  %\  after  reversion, 

X  ^  c^z  Ar  c^z^  +  c^z^  +  .  .  .  =  c^{y  -  a,)  +  c^{y  -  a^Y  +  .  .  . 

Example  1.  Effect  the  reversion  of  y  =  x  —  ^x^  -{- M^  -  4a^  +  .  .  . 
Assume  x  =  ay  -{-  by"^  -\-  cy^  -\-  dy^  +  •  •  •      SnLsMtuting  the  value  of  y, 
we  get 

x  =  a{x-  2a2  +  Sa;^  -. ,.)  +  h{x-2x''  +...)'  +  c(a;- 2a;^  +  .  ,f  -\-d{x-.. .)*+... 
=zax-\-{h-  2a)x''  +  (c  -  4&  +  da)x^  +  (tZ  -  6c  +  106  •-  Vi^  +  .  .  . 

We  find  Id  succession  that  a  =  1,  b  =  2,  c  =  6,  d  -  11      Uence 
i 

x  =  y  +  2y'-{-5y'-}-Uy'  +  ... 

Example  2.  Find  a  series  in  y  that  will  give  a  solution  ol  x"^  —  2x  -{-  S  =  y. 
Setting  y  —  3  =  2,  we  have  z  =  x^  —  2x.     Hence  z  change?  in  sign  when  x 
does.     Hence  the  required  series  for  x  contains  no  even  powers  n*  z-     Ket 

X  =^  az  -\- b^  -\-  c^  -\-  dz'  -\-  ,  .  . 

.'.    X  =  a{-2x-\-x^)  -f  b{-2x  +  a^f  +  c{-  2x -\- x^f -^  d{-2x  +  x")  -J- 

=  -  2aa;  +  (a  -  Sb)x'  +  {12b  -  S2c)x^  +  (  -  6&  +  80c  -  128d)x'^  +  . 

1       ,        -1  -3      ^       -3 

•••     ^=-T'     ^^16"'     ^  =  T28'     ^  =  ^'-- 

.-.     x  =  -  i{y  -  3)  -  j\{y  -  Sf  -  jUv  -  ^f  -  ^U(y  -  3)'  +  .  . . 

The  solution  x  is  valid  only  for  values  of  y  for  which  the  series  is  convergent 
The  fact  that  the  coefficients  of  the  even  powers  of  z  are  zero  may  also  be 
verified  by  computing  them  directly,  as  was  done  for  the  odd  powers. 


Sec.  123]  COLLEGE  ALGEBRA.  129 

EXERCISES. 
Expand  to  five  terms  in  ascending  powers  of  x  : 


«•   .    .'^"^."^   .»  6.  ,_}^Z^  2  •      7.  (1  -  x)\  8.  (1  +  ic)-  '. 

Find  the  coefficient  of  the  general  term  in  the  series  for 


'  {\-x){l-yx^)'  '  l  +  3uj  +  2.c2'  '  x^x^'  '  x' -\- X 

Obtain  the  reversion  of  eacli  of  the  series 

IZ.  y  =  X  -  X'  -^  x^  -  x^  -^  .  ,  .  14.  3/  =  ^  +  Y+  -^  -f  y  -h  .  .  . 

Solve  by  series  the  equations 
lb.  y  =  x  -  xK        16.  y  =  1  -h  "ix'  -f  ^x^  -h  6a*.         Vl.  y  ^\'\  x-\-  a» 


CHAPTER  XV. 
BINOMIAL  THEOREM    FOR  ANY   INDEX.* 

124.  Consider  the  power  series  in  x 

/.x  .  .        .  nin—l)  .,          ,   n(7i—l){n—%),..(n—r-\-l)  ^, 
(1)  i+nx+  -^--^x^-\.,  . .+  -^ \,2\.,.r ^^^  +  ' 

If  n  is  a  positive  integer,  the  series  is  a  finite  series  terminating  | 
with  the  n  +  1st  term,  since  the  {n  +  2)nd  and  all  subsequent  terms  ; 
contain  the  factor  n  —  n.  In  this  case  the  series  is  the  expansion  : 
of  the  binomial  (1  -\-xY  [see  the  proofs  in  §§  91  and  99]. 

When  n  is  not  a  positive  integer,  the  series  is  an  infinite  series, 
since  no  one  of  the  factors  ?^  —  1,  ^  —  2,  .  .  .  can  now  be  zero.  The  | 
first  step  in  the  study  of  the  series  is  consequently  to  determine  for 
what  values  of  x^  if  any,  the  infinite  series  is  convergent.  We 
employ  the  ratio  tests  of  §§  116  and  117.  The  (r  +  l)st  term  u^^i 
is  exhibited  in  the  series  (1).     The  rth  term  is  consequently 

_  n{n  —  \)(n  —  2) , . ,  (n  -  r +  2)     _, 

Ur  — ^^ 


1-2-3...  (r- 1) 

Ur  \         r        J  \  r 

Hence  this  ratio  approaches  —  a:  as  r  increases  without  bound.  Its 
absolute  value  will  ultimately  become  greater  than  unity  if  |a;|  >  1, 
so  that  the  series  is  divergent  in  that  case  (§117).  But  for  |  a:  |  <  1, 
the  absolute  value  of  the  ratio  will  be  less  than  some  fixed  quantity 
which  is  itself  less  than  unity'  for  all  values  of  r  which   exceed 

*The  theorem  was  discovered  by  Sir  Isaac  Newton.     The  proof  given 
is  a  modification  of  that  due  to  Euler. 

130 


;  Sec.  125]  COLLEGE  ALGEBRA,  13I 

\  a  certain  number  and  therefore  the  series  is  convergent  (§116). 
\  For  example,  if  n  =  6|,  a;  =  —  -i-,  tlie  ratio  will  be  a  positive  frac- 
tion for  r  ^  3.      [It  may  be  shown*  tliat  the  series  is  convergent 

for  re  =  1  provided  u  <  —  1,  and  for  x  =  ■—  1  provided  71  >  0.] 
The  series  (1)  is  convergent  if  x  is  numerically  less  than  unity ^ 

hut  is  divergent  if  x  is  numerically  greater  than  unity, 

125.   Suppose  that  x  has  a  fixed  value  which  is  numerically  less 

than  unity.   -  The  series  (1)  has  a  finite  sum  whose  value  depends 

upon  7^;  we  denote  the  sum  by/(yi).     Then  the  series 

(2)  l+m:r+     ^^^   V  +  .  .  .  +  -^ 1.2.., r  ""  +  •  '  • 

will  have  the  finite  sum  f{ni).  If  the  series  (1)  and  (2)  be  multi- 
plied together  and  the  product  be  arranged  according  to  ascending 
powers  of  x,  the  coefficient  of  x  will  evidently  be  n  +  ni  and  the 
coefficient  of  x^  will  be 

n(n  —  1)               ,    mim  —  1)       (?^  +  m)  (71  -\-  m  —  1) 
A__  +  nm  +  —3-^-  = ^^ . 

Likewise,  the  coefficient  of  x^  is  seen  to  equal 

{n  +  m){7i  -\-  m  —  l){n  -{-  m  —  2) 
1.2.3'  • 

As  far  as  the  first  four  terms,  the  product-series  is  of  the  form  (1) 
when  n  is  replaced  by  7i -\- m,  giving  series  (4).  We  proceed  to 
prove  that  this  result  holds  true  for  all  the  terms  of  the  product- 
series.  This  is  readily  proved  for  the  case  in  which  7i  and  m  are 
any  positive  integers.  For,  in  this  case,  the  series  (1)  is  the  expan- 
sion of  (1  +  xY  and  series  (2)  is  the  expansion  of  (1  +  x)"",  so  that 
the  product  is  (1  -f  :r)''  +  "',  whose  expansion  is  the  series  (4),  since 
^  +  m  is  a  positive  integer.    Hence,  if  71  and  m  arc  positive  integers, 

(3)  f{7i)  xf{77i)=f{n  +  ni). 

*  Charles  Smith,  Treatise  on  Algebra,  Art.  338. 


132  BINOMIAL    THEOREM  FOR  ANY  INDEX,  [Ch.  XV 

Since  n  and  m  are  arbitrary  positive  integers,  they  must  be  rep- 
resented by  general  letters,  the  restriction  to  positive  integers 
being  supplementary  and  not  expressed  in  the  notation  for  7iand  m. 
On  account  of  the  generality  of  the  notation  for  n  and  m,  the  way 
in  which  the  coefficients  of  the  series  (1)  and  (2)  combine  to  give 
the  coefficients  of  the  product-series  must  have  been  2^  formal  process 
independent  of  the  supplementary  restriction  that  m  and  n  are  posi- 
tive integers.  The  argument  is  based  upon  a  principle  known  as 
"the  permanence  of  form.'^  Its  validity  in  the  present  case  is 
established  in  §  126.  Hence  the  form  of  the  coefficients  of  the 
product-series  remains  the  same  when  that  restriction  is  removed, 
so  that  the  product-series  is 

/>«\      ^/      I      \       1    I    /      I      N     1    {n -\- m)(n -\r  m —\)  „   , 
(^)     /{^  +  ?«)==  1  +  (^  +  m^  +  ^—^ — '-\^ ^-x^  +  . . . 

whatever  the  values  of  n  and  m  may  be.  Since  |a;  j  <  1,  the  latter 
series  is  convergent.  Formula  (3)  therefore  expresses  the  relation 
between  the  finite  sums  of  the  three  series  (1),  (2),  and  (4). 

126.  To  give  an  explicit  proof  of  formula  (3),  consider  the 
coefficient  of  x"^  in  the  product-series,  r  being  a  fixed  positive 
integer.  It  is  clearly  the  sum  ^  of  r  +  1  terms,  one  of  which  is 
the  coefficient  of  x""  in  series  (1),  and  another  the  coefficient  of  x^ 
in  series  (2).     We  wish  to  prove  that  this  sum  '^  equals 

(n  +  m){n  -{-m  —  \) . .  .  {n-\- m,  —  r  -\-\) 
^^  1.2.  .  .r  • 

Multiplying  both  ^  and /by  r!,  the  equation  to  be  proved  is* 
n(n  —  1) . . .  (7i  —  r  +  1)  -f  rmn(n  —  1) .  .  .  (^^  —  r  +  2) 

+  . . .  +  m{m  —  1) ...  (m  —  r  +1) 
=:  (n  -f  m){n  -^m  —  \) . ,  .  {n-\-  m.  —  r  •\-  1), 

*  Denoting  for  brevity  the  product  p(p  —  1)(7?  —  2).  .  .  (;?  —  «-f-l)  byp» 
the  relation  is 

^r  +  rCi  nr-\m[  +  rCg  Tlr  -2  ^2  +  •  •  •  +  r^r  -  i  W,  W,,  _  ,  '-|-  W^  =  {n  +  m)r 

and  is  known  as  Vandermonde's  Theorem.     Compare  Ex.  14,  p.  99. 


Sec.  1271  COLLEGE  ALGEBRA  133 

\  in  each  term  of  which  n  and  771  enter  to  the  degree  r.     Now  the  re- 

'  lation  has  been  established  for  the  case  in  which  n  and  m  are  both 

positive  integers.     Let  7i  be  an  arbitrary  quantity  and  let  the  rela* 

tion  be  expanded  and  the  terms  arranged  according  to  descending 

I  powers  of  71,     We  obtain  an  equation  of  the  rth  degree  in  71,     Let 

\  m  be  any  particular  positive  integer.    Then  the  equation  is  satisfied 

'  by  an  infinite  number  of  values  of  71,  namely,  the  positive  integers. 

Hence  (§  78)  the  equation  is  an  identity  in  71  and  is  satisfied  by  every 

value  of  71.     Hence  the  relation  is  established  for  71  arbitrary  and  ?/i 

any  positive  integer. 

Consider  the  original  relation  for  any  particular  value  of  72, 
whether  or  not  a  positive  integer.  We  expand  the  members  of  the 
relation  and  arrange  the  terms  according  to  the  powers  of  m, 
obtaining  an  equation  of  the  rth  degree  in  m»  By  the  previous 
case,  this  equation  is  satisfied  by  all  positive  integers  m,  whose 
number  exceeds  r,  and  is  consequently  satisfied  by  every  value  of  m. 
Hence  the  relation  is  true  for  71  and  m  both  arbitrary  quantities. 

It  follows  that  -2"  =/and  hence  that  the  product-series  is  iden- 
tical with  the  series  (4). 

127.  By  repeated  application  of  (3),  we  find  that 

f{n)  XM)  Xf{p)X...  Xf{q)  =f{7i  +  77i+p+,  .  .  +  q). 

Let  there  be  5  terms  n,  ?/?,  p,  .  ,  , ,  q  and  take  each  equal  to  the 
fraction  r/s,  where  r  and  ^  are  positive  integers.     Then 

w:-)r=<-+^+--+r)=/ex')=/«- 

Since  r  is  a  positive  integer,  f{r)  =  (1  -j-  xy.     Hence 

{/(:t)P=(i  +  -)-   /(It)  =  (!+-)'• 
Hence^  ifn  is  a  positive  fraction  -  ,  series  (1)  equals  (1  -f-  x)s. 


134  BINOMIAL    THEOREM  FOR  ANY  INDEX.  [Ch.  XV 

In  the  identity  (3)  put  n  =  —  m.     Then 

/(»Ox/(-«0=/(o)  =  i. 

•■••^^-"^=/lk  =  (rT-^'  =  ^'  +  ^"^"""' 

if  m  is  a  positive  fraction.     Hence  if  n  is  a  negative  fraction,  series 
(1)  equals  (1  -|-  xy.     We  may  now  state  the  final  theorem: 

If  X  is  in  absolute  value  less  tlia7i  ^mity,  and  if  n  is  any  rational 
7Uimher^  7ve  have  the  expansion 

...«.,          ,    n(?i  —  1)    ^   ,   7ihi  —  l)(n  —  2)  „      ' 
(5)     {l-\-xr=^l  +  71X  +  -A___Z^2  ^  _A _|V_ _^^3_j_  ^  ^  ^ 

This  result  is  known  as  the  Binomial  Theorem. 

Since  any  real  number  may  be  approximated  to  any  desired 
degree  of  approximation  by  a  rational  number,  we  can  establish,  by  a 
limiting  process  (§  16),  the  Binomial  Theorem  for  any  real  index  7i, 

128    Example  1.  Expand  (1  -f  ir)-iby  the  binomial  theorem. 
If  I  oj  I  <  1,  formula  (5)  gives,  for  n  =  —  1,  the  result 

=  1  -X  +  X''  -  x^  -i-  .  .  .  -i-  (-  lYxr  +  .     . 
Tlie  resulting  series  is  an  infinite  geometrical  progression  with  the  ratio 

-X.     Hence  (§69)  its  sum  is  - — ■ — when  Icri  <  1. 

vo        /  1  -^  X  II 

Example  2.  Expand  4/I  —  ^  by  the  binomial  theorem. 
Setting  —  y  rrr  X,  n  =  {,  formula  (5)  gives,  if  |^|  <  1, 

(i-^)^  =  i  +  i(-^)  +  M^\-^)^+...+iiti^^  ,. 

r. -i       ^         12  l-3-5...(2r-3)  ^ 

•.  |/1  -  2^  =:  1  -  ty  -  4/  -  . . . ^^^ y^  -  . .  . 

This  result  agrees  with  that  obtained  by  undetermined  coefficients  (§  122^. 
Example  3.     Find  ^99  to  six  decimal  places. 


upon   setting  y    ^    .01    in  the  formula  of  Ex.    2.     .-.  i/99  =:  9.949874 -f. 
The  terras  beyond  the  fourth  do  not  affect  the  first  eight  decimal  places. 


Sec.  128] 


COLLEGE  ALGEBRA, 


135 


Example  4.  Find   1^66  to  six  decimal  places. 


=  4(1  +  .0104166  -  .0001085 


^  3     32 


1/iy     5/n«_     n 

9i^32j  "^81\32y       •  •  ■ 
.0000019  -  .  .)  =  4.0412400. 


EXERCISES. 

X  Expand  to  four  terms  and  give  the  (r 
1    {l-\-x)-"\         2.  (1  -  x)-K  3.  (1 

5.    |/1~^^2^^.  6.    7/-r-f^-  7.  x\l  -  x'i 


l)st  term: 

3a;)^.  4.  (2  -  ^x)K 

'=^'^  8.    \/^  -  x^. 


Find  to  five  decimal  places  the  value  of 

9.    \/ll.  10.   ^"12(5.  11.    ^1002.  12.  f  3r2a 

13.  Find  the  coefficients  of  x^  and  x^  in  (1  +  2.?;  -|-  %x^)-'^. 

2  —  x^ 

15.  Find  the  coefficients  of  x^»\  a-^'"  - 1,  a;^"*  +  1  in  (1  +  a;  +  x^)  -  *. 

16.  The  coefficient  of  x^  in  (1  —  a;  -f  x^  —  x^  -^  x^)-^  is  zero. 

17.  The  coefficients  of  x^  and  x'  in  (1  +  a;-V  ^^  -\-x^)~^  are  zero. 

18.  Prove  that  (1  -  ic)-2  =  1  +  2icH-3aj2  -f  . .  .  +  [r-^l)x^-\- . » o 


14.  Find  the  coefficient  of  x'^  in  the  expansion  of 


CHAPTER   XVL 
EXPONENTIAL   AND   LOGARITHMIC   SERIES. 
129.  If  ?^  >  1,  we  have,  by  the  Binomial  Theorem, 

/     ,    1 V'"^.  -I    I   ^^   I    '^^^{^^^  ~  -^)    I   ^^^f^^^'  ~  1)(^^^  ""  ^)    I 

-1  +  ^+  2!  "^  3!  +• 

For  the  case  a:  =  1,  we  have 

V  +  n)   -1  +  1^ 2! + 3! +••■ 

.    1  _L  ^  J.  ^^^  ~  ^/^)  _L  a--(a^  -  l/w)(a;  -  2/w) 
•  •   1  +  a;  H ^1 1 — 3I H  •  •  • 


{>  +  '  +  <^^  +  -(' 


a  result  true  for  any  value  of  /j  >  1.     Taking  the  limit  as  71  in- 
creases without  bound,  we  get 

l  +  =r  +  2-i  +  3T+-..=(l  +  l  +  2T  +  3!+---)- 

The  general  term  of  the  series  on  the  left  is 

x"-        ,.    .^  x(x  -  l/n){x  -  2/n)  .  ,  .  \x  -  {r  -  l)/n} 

—r  =  limit .   ,^  o • 

r!       ^  .  ^  1-2-3  .  .  .  r 

n  —  CO 

136 


Sec.  129]  COLLEGE  ALGEBR/I,  I37 

For  small  values  of  r  this  evaluation  of  the  limit  is  evident.  To 
prove  that  the  result  holds  true  when  r  becomes  large,  we  denote  by 
Ur  the  fraction  whose  limit  we  seek.     Then 


u. 


I 


'^    *  r  \r       n    ^  ml 

,\    limit  Ur  —  -  limit  «^r-i« 
r      ' 
n^co  n=co 

Oj2  /yi3 

But  the  limit  of  w„  was  seen  to  be  —r;  hence  the  limit  of  ^„  is  ttt. 
^  2 !  ^      3 ! 

By  induction,  it  follows  that  the  limit  of  u^.  is  x^'/rl     Set 

(1)  ,Hl  +  H-^  +  ^  +  ^+...+^+-.. 
The  above  result  may  now  be  written 

(2)  ^^i  +  :r.  +  lj  +  l^  +  ...+^+... 

To  evaluate  e,  notice  that  jj  is  one-fourth  of  ^,  etc.     Hence 

e  =  2+  .5 +  .1666 +  .0417 +  .0083 +  .0014  +  .0002  +  .  .. 
Hence  e  =  2.7182  to  four  decimal  places.     To  ten  places,  we  get 

(3)  e  =  2.7182818284. 

We  may  now  expand  a^  into  a  power  series  in  y.     Since 

a  =  e^^'^e^,     ay  =  e^  ^«^A 

by  the   third   law   of   indices.      Putting  x  =  y  log^a,    series   (2) 
gives 

(4)  a^  =  1  +  y  log,a  +  l^y^log^af  +  ~y\\ogsf  +  •  •  • 

This  result  is  known  as  the  Exponential  Theorem.    The  relation  (2) 
is  valid  for  any  a:,  and  (4)  for  any  y. 


138  EXPONENTIAL  AND  LOGARITHMIC  SERIES,       [Ch.  XVI 

130.  We  may  give  a  second  proof  of  the  result  (2),  following 
the  method  employed  for  the  Binomial  Theorem  (Chapter  XV). 
Set 

i^(^)  =  1  +  o:  +  |_r  + -gj  +  .  .  .  +^  +  .  .  ., 

For  every  value  of  x  and  z^  these  series  are  convergent  (§  116). 
Form  the  product  of  the  series  F{pc)  and  F{z),  The  coefficient  of 
x^'z^  in  F{x)  •  F(z)  is  1/  (r!^!).  But  x^'z^  occurs  in  the  series 
F{x  -f-  2;)  only  in  the  expansion  of  the  term  {x  -^  z)^^^  -~  {r  -\-  s)  ! 
and  occurs  there  with  the  coefficient  (as  shown  by  the  Binomial 
Theorem  for  positive  integral  index) 

1  (r+5)!  __      1 


(r+  5)1         r\s\  r\  sV 

Hence  the  series  obtained  as  the  product  of  the  series  F(x)  and 
F{z)  is  identical  with  the  series  F(x  -\-  z)»  v 

.-.  F(x)-  F{z)  =  F{x  +  z), 

for  all  values  of  x  and  z.     It  follows  that 

F{x,)  .  F{x^)  .  F{x,)  .  .  .  F{x:)  =:  F{x^  +  x,  +  x^+...+  x,), 

for  any  positive  integer  7i  and  any  values  oi  x^,x^,  ,  ,  ,  ^  x^-  Taking 
the  particular  values  x^  =  x^=^ ,  .  .  =  2:^  =  1,  we  get 

\F{l)Y  =  F{n), 

But  F{1)  =  1  +  1  +  ^  +  .  .  .  =  e.     Hence  F{n)  =  e^  so  that 
(2)  is  proved  for  tlie  case  x  =  n,  ^  positive  integer.     Taking  next 


Sec.  IBIJ  COLLEGE  ALGEBRA.  I39 

the  particular  values  x^  =  x^=  ...  =  ir^  =  — ,  where  m  also  is  a 
positive  integer,  we  get 

By  the  case  just  established,  F{m)  —  e'"'.     Hence 

Hence  formula  (2)  is  proved  for  the  case  x  —  ^ — ,  a  positive  frac- 
tion. To  extend  the  proof  to  the  case  x  —  —f,  f  being  a  positive 
fraction,  we  note  that  F{-f)'F{f)  =  F{0)  =  1,  whence 

Hence   formula    (2)   holds   for  all  rational    values  of  x.     By  the 

method  of  limits  (§§  16,  105),  the   result  may  be  extended  to  any 

real  \alues  of  x. 

Ill 
131.  Example.     Find  the  sum  of  1  -f  -^-:  +  -j^  +  -^  +  •  •  •  «^  infinitum. 

For  X  =2  -\-  1  and  x  —  —  1,  formula  (2)  gives 

...   .  +  .-.  =  2  +  1  +  1,  +  !  +  ... 

The  required  sum  is  therefore  ^{e  -\-  e-  i). 

EXERCISES. 

1.  Show  that  ^,-1  =  1  +  I  +  1  +  A  +  .  .  , 

2.  Evaluate  a""  -  b'' +  ^(a'  -  b')  +  -{a'  —  5^)  +  . .  . 


3.  Expand  (e-^  +  e^^)  -^-  e^^  into  a  power  series  in  x, 

P        2=^        ^!   I    i' 
1 !   '    2!        31        4 ! ' 


4.  Prove  that  5^  =  t^  +  -^ ,  +  oi  +  x,  +  •  • 


MO  EXPONENTIAL  AND  LOGARITHMIC  SERIES,       [Ch.  XVI 

6.  Prove  that  1  =  — 4-  — -}-Ai_t_L,.. 
2!        3!        4!   '    5!  "^ 

12        o'-^         8^ 
6.  Prove  that  ^  =z  1  +  ^^+  ^  +  |j  +  .  .  . 

.     7.  Findthesumofl  +  ^^+^'  +  ^V.. 

132.  Theorem.     If   \x\  <1^  log^  (1  +  x)   equals  the   sum   of 
the  series 

/y»2  /y.3  /y.4  ^r 

(5)  ^_|  +  |-.^-+...._(_l)^?-+... 
Substituting  1  +  ^  for  a  in  the  equation  (4),  we  get 

(1  +  ^)i/  =  1  +  2^  log,(l  +:,)  +  ^^2]log,  (1  +  :r)  [2  +  .  .  . 
Also  by  the  Binomial  Theorem,  we  have,  when  |  a;  |  <  1, 

(i+.)«=i+,.+fcil.»+. .  .+^fcil^__(i^i±l),.+ . . . 

In  the  second  member  the  coefficient  of  y  is 

.+(ri.>..+...+(-')(-^^-(-+'v+..., 

which,  upon   simplification,  is  series  (5).     The  coefficient  of  y  in 
the  series   l-{-y  log^  {I  ~\-  x)  -\-  .  .  .  is  log^  (1  +  x).      Since  the 
two  series  are  equal  for  all  values  of  y^  we  may  equate  the  coeffi- 
cients of  like  powers  of  y  (§  121),  so  that  the  theorem  follows. 
By  equating  the  coefficients  of  ^^,  we  find,  similarly,  that 

illog.  (1  +  x)\^^  ix^  -  i(l  +  ^)x^  +  HI  +  i  +  \)x*  -... 

133.  Keplacing  a:  by  —  ic,  we  obtain  from  (5),  for  |  a;  |  <  1, 

/y2  /y«3  />«4  /y,r 

(6)  \og^{\-x)  =  -x---~---  ...---... 


» 


Sec.  134]  COLLEGE  ALGEBRA.  141 

Subtracting  from  (5)  and  using  log  -7-  ==  log  a  —  log  h,  we  get 

(7)  log4-±|  =  2(0.  +  I'  +  I'  +  y  +  . . .). 

Substituting  in  (7), 

'in  —  n         ^1^1  +  .'^       in 

X  = ; ,  so  that  -— ' —  —  --, 

m  -^  n\  1  —  X       n 

we  get,  for  any  positive  values  of  in  and  n, 

134.  7'able  of  logarithms  to  the  base  e  =  2.71828  +;  natural 
logarithms.     For  771  =  2,  n  =  1,  formula  (8)  gives 

log.  3 =2{i+i.(i)^+ia)^+i(i)^ +...}, 

from  which  we  get  log^  2  =  .693147  to  six  decimal  places. 
For  771  z=:  3,  71  =  2,  formula  (8)  gives 

log,  3  -  log,  2  =  2  li + mr + my  +  my + . . .  i 

=  2{.2  +  .002667  +  .000064  +  .000002} 
=  .405466     (to  six  decimal  places). 

By  carrying  the  work  to  more  places,  the  sixth  place  is  seen  to  be  5 
instead  of  6.     Adding  log,  2,  we  deduce  log«  3  =  1.098612. 

For  7n  =  5,  71  =  3,  formula  (8)  gives  log,  5  =  1.609438. 

We  may  now  deduce  log,  4  =  2  log,  2,  log,  6  =  log,  2  +  log,  3, 
log,  8  =  3  log,  2,  log,  9  =  2  log,  3,  log,  10  =  log,  2  +  log,  5.  We 
get  log,  7  from  (8).  We  thus  get  the  numbers  in  the  second  column 
of  the  following  table : 


I 


142 


EXPONENTIAL   AND  LOGARITHMIC  SERIES,        [Ch.  XVI 


N 

log,  N 

logio  ^ 

1 

0 

0 

2 

0.693147 

.301030 

3 

1.098612 

.477121 

4 

1.386294 

.602060 

5 

1.609438 

.698970 

6 

1.791759 

.778151 

7 

1.945910 

.845098 

8 

2.079442 

.903090 

9 

2.197225 

.954243 

10 

2.302585 

1.000000 

'  To  obtain  log^^  iV^,  we  multiply  log^  JVhy  [see  §  19] 
1  1 


logic 


.4342945. 


loge  10  2.302585 
Hence  the  numbers  in  the  third  column  of  the  table  are  derived 
from  the  corresponding  numbers  in  the  second  column  by  mul- 
tiplication by  the  constant  .4342945,  which  is  called  the  modulus 
of  common  logarithms  (namely,  logarithms  to  the  base  10).  Loga- 
rithms to  the  base  e  are  called  natural  logarithms  or  Napierian 
logarithms  (in  honor  of  their  discoverer,  Lord  Napier). 

135.  Interpolation.     For  m  =n  -\-  a  and  m  =  n  -\-  b,  formula 
(8)  gives 

^^^^  (^^+")-^^^^  "  =  M  2^  +  M2..+. 
log,  (,,+^)_log,  n  =  2  i  -1-  ^  1  (-—Y 


\2n+b'^  3  \2n+b/  "^  5 


\27i+b) 

If  a  is  relatively  small  compared  with  71,  the  3rd,  5th,  and  higher 
powers  of  a  -^  {27i  +  «)  may  be  neglected  for  approximate  results. 
Similarly  for  b.  Then,  approximately, 
(9)  |log  {n  +  a)  —\ogn\ }  :{log  {7i-\-b)  —  log  71}=  a  :  b, 
a        ^         b       _  a     271  -\-  b 


since 


2n  +  a   *    27i  +  b       b  '  2)1  +  a 


and    the    last   factor   differ 


from  1  by 


a  —  b 


2n  +  a 
hold  for  common  logarithms  base  10, 


If  (9)  holds  for  logarithms  base  e,  it  will  also 


Sec.  136]  COLLEGE  ALGEBRA,  143 

To  consider  a  numerical  example^  let  the  base  be  10  and  let 

n  =  4550,  a  =  2,  Z'  =  10.     Then  —-^^ —  =.  -^  z=  .00022  to  five 

2}i  4-  a       91u2 

decimal  places:  2^ — -^  =  — —-  =  .00110    to   five   decimal    places. 
^  2)1  +  0       911  ^ 

Their  cubes  may  be  safely  neglected.     By  a  Table  of  Logarithms, 

log  4550  =  3.65801,    log  4552  =  3.65820,     log  4560  =  3.65896. 
.  •.     log  4552  —  log  4550  =  .00019,  log  4560  —  log  4550  =  .00095. 

Hence  by  the  formulae  or  by  the  Table,  the  diiferences  of  the  loga- 
rithms are  approximately  proportional  to  a  :  ^,  viz.,  1:5,  so  that 
(9)  is  true.  To  pass  from  the  results  given  by  the  formulae 
(base  e)  to  those  given  by  the  Table  (base  10),  we  must  employ  as 
multiplier  the  modulus,  practically  .4343  in  value.  Multiplying 
log,  4552  -  log,  4550,  of  value  2  (.00022),  by  .4343,  we  get 
logio4552  -  logi,450,  of  value  .00019. 

For  relatively  small  differences  in  tlie  niimhers^  the  logarithm's 
increase  is  approximately  in  proportion  to  the  number^ s  increase, 

136.  Example  1.  Expand  loge(l  —  x  -{-  x^)  into  a  power  series. ' 

log  (1  +  ^  +  ^=^)  =  log  ^-^1  -  log  (1  +  ^)  -  log  (1  +  0^) 

r    i^\n  2 

The  coefficient  of  x^  is  if  n  is  not  divisible  by  3,  but  is  (  —  1)«  if 

n  n 

n  is  divisible  by  3. 

Example  2.  If  ex,  (5  are  the  roots  of  ax^  -\-  hx  ■■{-  c  —  0,  then 

log  {a  -hx^  c^')  =  log  ^  H ^^x i^^   '^ ^^   ~     • 

h  c 

By  §  32,  we  have  a  -f  /?  = ,  a6  =  — .     Hence 

a  —  hx  -\-  cx^  —  «(1  -f  ax>j{X  -\-  ftx) 
log  {a--bx  +  ex")  =  log  ^  +  {ax  -  Id'x'  +  •  •     )  \  {fix  -  |/5V  +  ...). 


144  EXPONENTIAL   AND  LOGARITHMIC  SERIES,       [Ch.  XVI 

Example  3.  Prove  that  e  is  incommensurable. 

Suppose  that  e  —  —,  m  and  n  being  positive  integers.     Then 

n       ^  "^  ^  "^  21  "^  8i  "^  •  •  *  "^    wl  "^  (7i  +  1)1  "^ 
Multiply  both  members  by  n  ! 

•■•     '''^''-'^'■  =  '^''^''  +  ^l  +  (n  +  ln+%)^\n+l),nl^)in+Z)+-- 
The  series  beginning  with  l/(/i  -|-  1)  is  less  than 


71  +  1   '    (^  +  1)2   '    (7i  4-  1)3    '    •  •     -   n 

Hence  the  sum  of  the  series  lies  between  — ■ — -  and   -    and  is  therefore  a 

n  -\-  I  n 

proper  fraction.     Hence  would  the  integer  m{n  ~  1)!  equal  an  integer  plus  a 
fraction.     Since  this  is  impossible,  e  cannot  equal  a  fraction. 

[EXERCISES. 

1.  Find  the  general  term  of  the  expansion  of  loge  (1  -f  «  -f-  a?^  +  a?), 

2.  Find  the  general  term  of  the  expansion  of  loge  (1  —  %x  -\-  2x^). 

/7>2  /7>3  nA  n,1  nfi 

Z.liy  =  .-^+^-^+.       ,.then.  =  2,  +  |,  +  |-  +  ... 
4.  loge  (.  +  .)-  log.  (n-c)  =  3  (^+  ^+  ^+  ^^  +  .  .  .). 
6.  log.2  =  -^+^  +  ^+^  +  .,,. 

6.  logeS  =  4  +  r^+ Fr^+ 5^^  +  •  •  • 

r  +  s  2rs  1  /    3r«    \'       1  f    2rs   \* 

'•  ^"Se-y^^  -  :;^r^^,  +  -^[^_f^->j  +  -5\^^rf-^)  +  •  •  • 

g-l,     1     a!''-!  1     x>  -  1 

'•  '"S'  *  -  i"+l  "*"  3   («  +  1/  +  "3"  (a;  +  1)»  +  • 


CHAPTER  XVII. 
SUMMATION   OF  SERIES. 

137.  The  method  of  undetermined  coefficients  (Chapter  VIII) 
and  the  method  of  mathematical  induction  (Chapter  XI)  have  been 
employed  to  effect  the  summation  of  certain  classes  of  series.  Like- 
wise, the  sum  of  an  arithmetical  or  a  geometrical  progression  has 
been  obtained  (Chapter  VI).  We  here  discuss  methods  of  greater 
generality  for  finding  the  sum  of  a  series 

^1   +   «^2   +  ^'3   +    •    •    •    +    ^n   +    •    .    . 

In  case  the  series  is  an  infinite  series,  the  result  is  valid  only  when 
the  series  is  convergent. 

For  an  arithmetical  progression  of  common  difference  d,  we 
;  have  u^_i  —  ti.^^  —  d=iu^—  n^_i ,  whence  ii,,  —  2?^„_i  +  w„_2  =  0. 
I  The  latter  relation  is  called  the  generating  relation,  since  the  series 
may  be  constructed  by  means  of  it.  Thus,  taking  n  =  3  and  4,  we 
get  W3  =  2ii^  —  tc^ ,  v^  =  22/3  —  i(^  =  3if^  —  2u^,  The  series  is  there- 
fore u^ ,  u^ ,  2u^  —  u^ ,  3ii^  —  2Wj , .  .  .  ,  of  common  difference  u^  —  n^. 

For  a  geometrical  progression,  the  generating  relation  is 
w„  —  rUn-i  =  0,  r  being  the  common  ratio.  Inversely,  every  two- 
term  generating  relation  leads  to  a  geometrical  progression. 

138.  In  general,  a  series  ic^  -\-  ii^-\-  u^~{-  .  .  ,  is  called  a  recur- 
ring series  of  order  Jc,  if  it  has  a  generating  relation  of  order  ^, 

tin  +  Oi'^n-l+  ^2^n-2  -f-   •   •    •    +  OjcVn-k  =0      {c^,   .   ,    .   ,   Cj,  COUStauts), 


I4<^  SUMMATION  OF  SERIES.  [Ch.  XVII 


I 


which  holds  iTue  for  n  —  Jc  -\-l,  h  ^2^  ,  .  .  ,  on  to  the  end  of  the 
series  when  the  latter  is  finite,  but  holds  ad  infinitum  when  the 
series  is  infinite.     • 

In  addition  to  the  above  examples,  consider  the  series 

Since  it  is  not  a  geometrical  progression,  the  generating  relation,  if 
one  exists,  must  be  of  order  at  least  two.  Trying  the  value  Ic  —  2, 
we  have  u^  +  c^Un-i  +  c^^(^n-2  =  0  for  ^^  =  3,  4,  .  .  .     Hence 

nx^-^  +  c^{n  —  l)x''-^  +  c^{n  —  2)x''-^  =  0. 
.-.  n{x^  +  c^x  +  cj  —  c^x  —  2c^  =  0    (for  7i  =  3,  4,  5,  .  .  .). 

.-.  x^  +  c^x  +  c^=0,         -  c^x  -2c^=0  (§  78). 

•       .         C|     —  /ClXj  Cn      —     X     » 

Hence  there  is  a  generating  relation  u^  —  2xu^_^  +  ^^^n-2  =  0  of 
order  two.  But  this  generating  relation  determines  the  above 
special  series  only  when  it  is  also  given  that  u^  =  1,  ii^  =  2x.  In 
fact,  the  relation  only  expresses  ^^g ,  ^^ ,  .  .  .  in  terms  of  u^  and  v^, 

139.  In  general,  a  generating  relation  of  order  Tc  determines  a 
particular  series  only  when  the  first  Jc  terms  of  the  series  are  given. 
On  the  other  hand,  if  it  is  known  that  a  series  has  a  generating 
relation  of  order  ^,  but  the  constants  c^^  c^,  .  ,  .  ^  c^^  are  not 
known,  we  must  have,  in  addition  to  the  first  h  terms  of  the  series, 
the  next  h  terms  in  order  to  determine  c^,  .  .  .  ,  6V 

If  the  general  term  is  given,  the  preceding  conditions  are 
evidently  satisfied  and  the  generating  relation  is  found  as  in  the 
above  example. 

Example.     Find  the  generating  relation  of  the  recurring  series 
1  +  ic  +  aj2  +  2a.'3  -4-  3^^  -f  2x^  +  2^«  +  7ic'  +    . 

If  the  generating  relation  were  of  order  two,  Un-\-  CiU^  -i-j-c^Un-z  —  0, 
we  would  have  x^-{-  CiX-{-  c^  —  0,  2x^  -f  CiX^  +  ^2^  =  0>  whence  x  =  0.  Assmn- 
ing,  therefore,  that  the  generating  relation  is  of  order  three, 

Un-i-CiUn-l-{~  Cu1tn-2-\-  C^Un-S  =  0, 


I 


Sec.  140]  COLLEGE  ALGEBRA,  I47 

we  have,  for  n  =  4,  5,  6,  the  conditions 

CiX^  -\-  CoX  -Y  Cz  —  —  2a^, 
2ci  a^  -\-  c-iX^  -\-  CzX  —  —  Zx^, 

Hence  Cx  —  —  x,  c^  =  Scc^  Cs  =  —  Zx^.     The  conditions  for  n  =  1  and  8  are 
seen  to  be  satisfied.     Hence  Un  —  xiin  - 1  -j-  2x^Un  -  2  —  Z^Un  _ 3  —  0. 

140.  Problem.      To  find  the  sum  of  n  terms  of  a  recurring  series 
of  order  ttuo, 

Let  the  generating  relation  be  n^  -\-  pxu.^_i  -^  qxhi,^_o  —  0.     Then 
pxSn  =  pa^x  +pa^x^  -f-  .  .  .  4-  pa,,_^x''-^  +  pa,,_^x'', 

qX^Sn  =  q%X^  +   •    .    .    +  5'^n  -3^''"^  +  qCln~'i^''+  qC^n-l  ^""^  *• 

Since 

a^x^-\-2M^x^+q%x^^0,  ,  .  .  ,  a,,_^x''--''-\-pa,,_>iX'''-^-^qa^_^x"'-^  —  0, 
Sn+px8n-\-qx^Sn=a^-^a^x-^pa^x-]rpa^^^x''-\-qa^_^Qf+qa,,_^x''+^ 

.    ^  ^  ^0  +  {(^i'\-P%)x  +  {pcin-i+  qctn-2)x''  +  qa,,_iX''+^ 
'  *      "  1  -{-px  -\-  qx^ 

t  2:  be  a  value  for  which  the  series  is  convergent,  so  that  (end  of 
§107) 

limit     aj^_iX''~'^  =  0. 

Then  a.^^^nf'  and  a„_ia;'*+^  will  also  approach  the  limit  zero.    Hence 

_  ^0  +  (^1  +  ya^x 
"^  \^px-\-qx^    ' 

For  a  recurring  series  of  order  one  or  tliree  we  find,  in  a  similar 
manner,  that  the  sum  to  infinity  is,  respectively, 

^0  ^0  +  K  +  ;^^o)^  +  (^2  +  P^x  +IfO^^ 


\-\-  px"*  \-\-  yx-\-  qx^  +  rx^ 


148  SUMMATION  OF  SERIES,  [Ch.  XVIL 

the  generating  relation  being,  respectively, 

Ur,  +  pxu^_^  =  0,  u^  -\-pxnn-i  +  qxhi^_2  +  rxH^_^  =  0. 

For  the  series  of  §  138,  we  have  ^o  =  1,  «i  =  2,  p  =  —  2,  5'  =  1. 

For  the  series  of  §  139,  ao  =  1,  ai  =  1,  a^  =  1,  p  =  —  1,  q  =  2,  r  =  —  S, 

1  +  2a;2 


••      ^OD 


1  -  aj  +  2a;^  -  Sx^' 


141.  The  fraction  obtained  as  the  sum  to  infinity  of  a  recurring 
series  is  called  the  generating  fraction,  since  the  series  may  be  ob- 
tained by  the  expansion  of  the  fraction  (§  122). 

When  the  generating  fraction  can  be  expressed  in  terms  of 
partial  fractions,  whose  denominators  are  powers  of  binomial  ex- 
pressions, the  general  term  of  the  recurring  series  may  be  readily 
obtained  by  the  binomial  theorem. 

For  example,  the  series  1  +  9:r  —  Wx"^  -\-  57a^  —  159aj^  +  •  •  •  has  the  gen- 
erating relation  Vn-\-  2xtin-i  —  Sx'^nn-2  =  0,  so  that  (§§  140,  80) 

1  +  llaJ  3  2 


l-\-2x-3x:'       1  -  X        1  4-  3« 

=  3(1  +  ic  +  aj2+  .  .  .  +  aj^+  .  .  .)-  2(1  -  dx -}- 9x^- \-(-^ Sxy  J^ . .  ,), 

in  which  the  coefficient  of  aj''  is  3  —  2(—  3)**. 

EXERCISES. 

Find  the  generating  relation,  the  generating  fraction,  and  the  coefficient 
of  aj**,  of  the  recurring  series: 

1.  1  +  4a?  -  2aj'  +  lOa;^  -  14a^  +  .  .  . 

2.  1  _  5a;  +  34a;2  -  260a;3  _|_  2056a;*  -  .  .  . 

3.  5  -  8a;  -h  66x'  -  17Qx^  +  800a;*  +  .  .  . 

4.  1  4.  6a;  -  4a;*^  -  40a;3  -  112a;*  -  32a;5  +  704a!«  4- .  .  . 

5.  1  -  5a;  +  9a;-  -  13ar^  +  .  .  .  6.  2  -f  3a;  +  5a''^  +  9ar»  +  .  .  . 
7.  4  -  5a;  +  7a;2  _  lla;3  _^  _  .  8.  1  +  7a;  -  a;^  -f  43a;»  +  .  .  . 
9.  la  +  2«  -f  32  +  .  .  .  +  7i2,             10.  18  +  2«  +  .  .  .  -f  n\ 


Sec.  142]  COLLEGE  ALGEBRA.  149 

THE  METHOD   OF    DIFFERENCES. 

142.  If  the  first  term  of  any  series  is  subtracted  from  the  second 
term,  the  second  from  the  third,  etc.,  a  series  is  obtained  which  is 
called  the  series  of  the  first  order  of  differences.  Proceeding  simi- 
larly with  the  latter  series  we  obtain  the  series  of  the  second  order 
of  differences  of  the  original  series.  Similarly,  we  obtain  the  series 
of  any  order  of  differences. 

Series:  w^  u^,  u^,  w^, .  . . 

1st   order  differences:     u.^  —  u^,  u^  —  ii^,  u^  —  u^,  ,  , , 

2nd  order  differences:  ii^  —  2u^-{-n^,       ^^  — ^^^+^^2'  •  •  • 

3rd  order  differences:  u^  —  3n^-{-^if,^  —  i{^,  ... 

Denote  hj  D^,  D^,  D^,  .  .  .  ,  the  first  terms  of  the  series  of  the 
1st,  2nd,  3rd,  .  .  .  ,  orders  of  differences,  respectively: 

A  =  ^^  -  '^^p  A  =  ^'3  -  -^'2  +  ^^'  A  =  ^^4  -  ^^^3  +  3^/2  -  ^^.  . . . 

We  observe  that  the  expressions  for  71^^  ti^,  n^  have  the  same  coef- 
ficients as  the  binomial  coefficients  in  (a  +  by,  {a  +  by,  (a  -\-  b)'^. 
We  proceed  to  prove  by  mathematical  induction  that 

(1)  «„  +  !  =  Ml  +  «A  +  — i:2~A  +---+nCrD,+  .-.+D„. 

Suppose  that  the  theorem  has  been  verified  for  any  series  as  far  as 
'the  (m  +  l)st  term  (as  has  been  done  for  m  =  1,  2,  3).     Hence 

(2)  u^^,=  n,+mD,+^C,D,+^C,D,+  .  .  .  +„,aA  +  .  •  -  +  I^m- 
Writing  the  analogous  formula  for  the  series  of  the  first  order  of 
differences  ii^—u^  =  D^,  u^—u^,  .  . ,  ,  u^j^i—u^,  ^^  +  2— '^m  +  i?  •  •  •  , 
the  first  terms  of  whose  1st,  2nd,  3rd, . . . ,  orders  of  differences  are 
i>2^  />3,  Z>^,  .  .  .  ,  respectively,  we  get 


150  SUMMATION  OF  SERIES.  [Oh.  XVII 

Adding  (2)  and  (3),  and  applying  ^(7^  +  ^(7. _i  =  ^  +  i(7^  (§89), 
we  get 

But  this  is  formula  (1)  for  the  case  n  =  in  -{-1.  Hence,  if  the  law 
holds  for  n  =  m,  it  holds  for  n  —  m  -{-  1.  But  the  law  holds  for 
n  =  1/2,  3.      It  therefore  holds  for  any  positive  integral  value  of  n. 

Example.  For  the  series  5,  8,  11,  14, ... ,  the  series  of  the  first  order  of 
differences  is  3,  3,  3,  ...  ;  that  of  the  second  order,  0,  0,  . .  .  Hence  D^  —  3, 
i>2  =  0,  i>3  =  0,  .  .  .  ,  i>r  —  0.  Hence  Un  +  i  =  5  -|-  3;^;  this  result  may  also 
be  derived  as  the  {ti  4-  l)st  term  of  an  A.  P.  of  common  difference  3. 

143.  Problem :   To  find  the  sum  S,,  of  the  first  71  terms  of  the 
seines 

(4)  v^,     v^,     t'g,     ?;,,...,  v^,  .  .  . 

Evidently  S^  is  the  {n  +  l)st  term  of  the  series 

(5)  0,     V^,     V^  +  l\,     %\  +  1^2  +  ^3'   •   •   •   '     ^i  +  ^'2  +  •   .   .  +  ^^n,  .   .   . 

• 
The  series  of  the  first  order  of  differences  of  (5)  is  clearly  (4). 
Hence  if  rZ^,  d.^,  d^, ,  .  .  ^  are  the  first  terms  of  the  series  of  the  1st, 
2nd,  3rd,  .  .  . ,  orders  of  differences  of  (4),  then  v^,  d^^  d^,  d^,  .  . .  , 
are  the  first  terms  of  the  series  of  the  1st,  2nd,  3rd,  4th,  .  .  .  ,  orders 
of  differences  of  (5).  Hence  the  {n  +  l)st  term  of  series  (5)  is -[by 
formula  (1),  with  21^  ^=0,  D^  =  v^,  D^  =  d^,  D^  =  J^, .  .  .] 

2) 


s  -nv  I  ^^^  -  ^^d  -4-  !^(!!_:zi_)(!^ 


-d,+  . 


+  ?i  ^'r  <^^r  -  1  +  •  •    •  +  ^4  -  !• 

144.  Example  1.  Find  the  Tith  term  and  the  sum  of  the  first  n  terms  of 
1,         15,        40,         79,     •   135,        211,  .  .  . 
1st  order  differences:     14,        25,         39,  56,  76,... 

2nd  order  differences:  11,         14,         17  20,  .  .  . 

3rd  order  differences:  3,  3,  3,  .  .  . 

4th  order  differences:  0,  0,  .  .  . 

Hence  the  ??th  term  and  the  sum  Sn  are  respectively 

1       -w  1      /O  *  O 


Sec.  144]  COLLEGE  ALGEBRA.  151 

,    n{n  -  1) , ,    ,    n{n  -  l){n  -  2) ,,    ,    n{n  -  l){n  -  2)(n  -  3)  ^ 

=  iin{?.ri'  +  26/^2  _l  69 ;i  -  74). 

Example  2.  Find  the  sum  of  the  squares  of  the  first  n  integers. 

Series:  1,     4,     9,     16,  ...  ,  r\{r  +  1)\  {r  -f  2)'^  (r  +  8)2,  .  .  . 

1st   order  differences:      3,     5,     7, ,  2r  +  1,  2r  +  3,  2r  +  5,  .  .  . 

2nd  order  differences:         2,      2, 2,  2,  .  . 

3rd  order  differences:  0, ,  0,  .  .  . 

,    v(n—  1)  ^  ■     nin  -  l)(?i  —  2)  _         7?   ,      ,    .,,_      ,    ,^ 

•■•   '^'^ "" ''  +  \  2    ^  "^  — r^"^3 —        T  ^''  +  ^^^^''  +  ^^* 

Example  3.  Find  the  number  of  shot  arranged  in  a  complete  pyramid 
^Yhose  base  is  an  equilateral  triangle. 

The  respective  layers  contain  the  following  number  of  shot: 

1st    order:     2,    3,    4, ,>•+!,  r  +  2,  r -f  3,  .  .  . 

Srul  order:        1,      1,  .  .  . .     ,1,  1,  .  ,  . 

3rd   order:  0, ,  0,  .  .  . 

EXERCISES. 

1.  Find  the  sum  of  the  first  7i  natural  numbers. 

2.  Find  the  ninth  term  and  the  sum  of  the  first  nine  terms  of  2,  9,  28,  65, 
126,  .  .  . 

Find  the  7ith  term  and  the  sum  of  the  first  n  terms  of 

3.  1,  8,  25,  52,  89,  .  .  .  4.  2,  12,  28,  50,  78,  112,  .  .  . 
5.  1,  11,  35,  79,  149,  251,  ...            6.  2,  20,  48,  86,  134,  192,  ..  . 

7.  1,  4,  10,  20,  35,  56,  .  .  .  8.  4,  8,  0,  -  32,  -  100,  -  216,  .  .  . 

9.  1^  2^  3=\  4^  .  .  .  10.  1*,  2\  3*,  4S  .  .  . 

11.  1-3,  4-7,  9-13,  16-21,  ..  .       12.  1-2,  2*4,  4-7,  8-11,  16-16,... 
Find  the  number  of  shot  in 

13.  A  pyramid  of  triangular  base  each  of  whose  sides  contains  20  shot. 

14.  A  pyramid  of  square  base,  15  shot  in  a  side. 

15.  A  truncated  pyramid  of  30  layers,  with  50  shot  in  a  side  of  the  square 
base. 

16.  A  truncated  pyramid,  whose  base  is  an  equilateral  triangle  with  100 
shot  in  a  side,  and  whose  top  has  56  shot  in  a  side. 

17.  A  complete  rectangular  pile  with  base  9  by  15  shot. 

18.  A  complete  rectangular  pile  with  15  layers  and  20  shot  in  the  longer 
side  of  the  base. 

19.  A  complete  rectangular  pile  of  71  layers  with  ?7i  shot  in  the  top  layer 
contains  ln{n  -f  l)(3m  -j-  2n  —  2)  shot. 


I 


CHAPTEE    XVIII. 

GRAPHIC   ALGEBRA; 

SIMULTANEOUS  EQUATIONS;   SIMULTANEOUS  INEQUALITIES; 

SOLUTION   OF   NUMERICAL  EQUATIONS. 


146.  The  numerical  results  of  an  investigation  are  often  ex- 
hibited in  a  table  of  statistics  accompanied  by  a  diagram.  For  a 
comparison  of  the  results,  the  diagram  shows  at  a  glance  what 
would  be  gotten  from  the  table  only  after  a  minute  study. 

Suppose  that  the  number  of  inches  of  rainfall  in  the  1st,  .  .  .  , 
12th  months  of  a  given  year  was  2,  1,  0.5,  3,  4,  3.5,  1,  0,  2,  1.5, 
0.5,  1  respectively.  We  may  represent  these  figures  graphically  by 
marking  off  the  proper  number  of  vertical  divisions  above  the  base 
line  at  the  point  indicating  the  corresponding  month  (Fig.  1). 


n 

I 

f 

v 

V 

p  . 

) 

\ 

\ 

\ 

/ 

\ 

\ 

<^ 

oL 

2    : 

— 

— 

—. 

J 

\ 

n 

)  11 

Suppose  that  a  quantity  y  varies  as  the  quantity  a;  in  such  a  way 
that  y  —  2x.  Then  for  a:  =  0,  ?/  =  0;  for  a;  =  1,  z/  =  2;  for  :c  =  2, 
^  =  4 ;  for  a:  =:  —  1,  «/  =:  —  2 ;  for  a;  =  —  2,  ^  =  --  4 ;  etc. 

We  may  represent  this  relation  graphically  as  in  Fig.  2,  in  which 
the  values  of  x  are  measured  along  the  horizontal  line  OX  (the 
positive  values  to  the  right  of  0  and  the  negative  values  to  the  left 
of  0)  and  the  corresponding  values  of  y  are  measured  in  the  vertical 

152 


Sec.  146] 


COLLEGE  ALGEBRA, 


153 


direction  (upwards  or  downwards,  according  as  the  value  is  positive 
or  negative).  It  is  easily  shown  that  the  points  so  located  all  lie  on 
a  straight  line  AOB  [see  §  151]. 

Y 


-f— 

1 

t_ 

1 

L 

/ 

0 

L 

1 

t 

1 

t 

Y' 
Fig.  2. 

146.  Definitions.*  The  two  perpendicular  lines  X'OX  and 
F' OF  are  called  the  a;- axis  and  the  ^-axis  respectively;  together 
they  are  the  axes  of  coordinates ;  their  intersection  0  is  called  the 
origin  of  coordinates.  To  plot  the  point  whose  coordinates  are  x 
and  y^  we  mark  off  x  units  on  the  ic-axis  from  the  origin  0,  and  on 
the  vertical  line  (parallel  to  the  y-axis)  through  the  point  thus 
reached  we  mark  off  y  units;   the  final  point  reached  is  designated 


(-2,4-3) 


(-2,^3) 


(-i-2»4-6) 


(  +  2,-3) 


Fig.  3. 


as  the  point  (x,  y).     In  Fig.  3,  we  have  plotted  four  points  whose 
pairs  of  coordinates  differ  only  in  their  signs. 

*  The  system  was  introduced  by  Descartes  in  1637. 


154  GRAPHIC  ALGEBRA;    SIMULTANEOUS  EQUATIONS.  [Ch.  XVIll 

Since  there  are  innumerable  pairs  of  values  x.,  y  which  satisfy 
the  equation  y  =  2x,  there  will  be  a  boundless  number  of  corre- 
sponding points  {x,  y)  which  lie  on  a  line,  called  the  graph  of  the 
equation  y  —  2x,  It  would  appear  from  Fig.  2  that  the  graph  of 
y  —  2x  is  a  straight  line  A  OB  [compare  §  151]. 

147.  The  graph  of  the  equation  x^  -f  ^^  _  95  will  be  a  circle  of 
radius  5  and  center  at  the  origin  0.  Indeed,  if  w^e  join  0  with  the 
plot  P  of  any  point  (x,  y)  such  that  x^  -\-  y'^  —  25,  OP  is  the 
hypothenuse  of  a  right-angled  triangle  whose  base  is  x  and  vertical 
side  is  y,  so  that  {OPf  —  x^-\-y^  —  26,  OP  =  5.  Hence  every  such 
point  Plies  at  the  distance  5  from  0.    The  circle  is  given  in  Fig.  4. 


\ 

Y 

R 

\ 

\ 

\ 

\ 

^ 

^- 

P 

/ 

^ 

/ 

\ 

\ 

/ 

\\ 

M 

Q 

0 

\ 

X 

V 

j 

\ 

\ 

/ 

\ 

/ 

V 

-^ 

^^ 

^ 

\, 

A 

The  graph  of  the  equation  2x-\-y  •=-\^  is  a  straight  line  (§  151 ). 
Hence  two  points  will  determine  it;  for  example,  (0, 10)  and  (5,  0), 
whose  plots  are  R  and  Q  in  Fig.  4. 

It  appears  from  the  figure  that  the  circle  and  the  straight  line 
RQ  intersect  in  the  points  P  and  ^,  the  ])lots  of  (3,  4)  and  (5,  0). 

The  corresponding  algebraic  problem  consists  in  determining  all 
sets  of  values  a;,  y  which  satisfy  both  of  the  equations 
x^^y'^  =  2h,     2a;  +  ?/  =  10. 


Sec.  148]  COLLEGE  ALGEBRA,  1 5  5 

AVo  must  therefore  consider  the  two  equations  to  be  simultaneous 
(§  34).     Setting  ^  =  10  —  "^x  in  the  first  equation,  we  get 

»x^-^x+lb  =  0,     x  =  ^  or  5. 
Hence  the  only  sets  of  solutions  are  ^  =  3,  y  =  4;  x  —  b,y  =  0, 

148.  In  general,  the  problem  to  solve  two  simultaneous  equa- 
tions in  x^  y  is  equivalent  to  the  problem  to  find  the  coordinates  of 
the  points  of  intersection  of  the  graphs  of  the  two  equations. 

In  particular,  two  simultaneous  equations  of  the  first  degree  in 

X  and  y  have  at  most  one  set  of  solutions  in  common.     This  result 

'*  follow^s  algebraically  from  the  solution  by  determinants  (§  35),  and 

geometrically  from  the  fact  that  the  two  graphs  are  straight  lines 

(§  151).     For  example,  the  simultaneous  equations 

2x-\-y  =:  10,     Ax—?,y  =  0, 

liave  only  the  common  set  of  solutions  x  =^  ^,  y  =  4;  their  graphs, 
the  straight  lines  EQand  OF  (Fig.  4),  have  only  the  common  point 
P,  viz.,  (3,  4). 

149.  If  we  attempt  to  solve  by  determinants  the  equations 

2x  +  y  z=  10,     2x  +  y  =  4=, 

we  get  O-o;  =  G,  O-y  —  —  12,  so  that  there  are  no  common  sets  of 
finite  solutions.  Plotting  the  graph  of  2x  -\-  y  =  4,  we  obtain,  a 
straight  line  wljich  apparently  does  not  intersect  the  graph  of 
2x  -{-  y  =  10.  The  fact  that  the  graphs  are  parallel  lines  may  be 
proved  by  observing  that  the  difference  between  the  ^-coordinates 
corresponding  to  the  same  ^-coordinate  is  always  6.  Hence  the 
graphs  have  no  point  of  intersection  in  the  finite  part  of  the  plane. 

EXERCISES. 

1.  Prove  that  the  graphs  of  x^  -f  ^2  _  25  ^j^^^  2x  -}-  y  —  1^125  are  tangent. 

2.  Do  the  graphs  of  x^'  -}-  y'^  —  25  and  2x  -\-  y  =  15  intersect  ? 

3.  Find  the  intersections  of  the  graphs  of  x^  -f  y^  =  25  and  x  -  5y  =  0. 

4.  Find  the  point  common  to  the  graphs  of  the  three  equations 

2x  -  Sy  =  7,     Sx  -  iy  =  13,     8^^  -  lly  =  33. 
6.  Find  the  interseciions  of  the  graphs  otx^  -\-  y"^  —  9  and  x  -{-  y  =  2. 


I 


156  GRAPHIC  ALGEBRA;    SIMULTANEOUS  EQUATIONS.  [Ch.  XVIII 
6.  Show  that  a  parallelogram  is  formed  by  the  graphs  of 

2. 


a'^  b         '     h^  a 


1,     -+^  =  2, 
a       b 


^4_  ^ 


150.    Theorem.      The  area  of  the  triangle  ivhose  vertices  are  the 
points  (x^,  y^,   (x^,  y^,  and  [x.^^  y^)  equals  the  determinant 


X, 

Vi 

i 

X, 

Vt 

i 

a;. 

Vz 

i 

Let  the  three  vertices  be  P, ,  P^^  P^ ,  respectively.     Evidently, 
triangle  P^P^P^  equals 
trapezium  Pj^j^gPg  —  trapezium  P^Q^Q^P^  —  trapezium  P^Q^Q^P^ 


But,  by  geometry,  the  area  of  a  trapezium  equals  one  half  the  sum 
of  the  two  parallel  sides  multiplied  by  the  perpendicular  distance 
between  them.  Since  x^  =  OQ^,x^  =  OQ^,we  have  x^  —  x^  ~  Q^Q^; 
similarly,  x^  —  x^^  Q^Q^ ,  x^  —  x^=  QiQr     Hence 

A  =  K^s  +  yi)(^s  -  ^'i)  -  ^{y,  +  ^i)(^.  -  ^1)  -  \{y,  +  y,){^z  -  ^,) 


Sec.  151J 


COLLEGE  ALGEBRA. 


157 


Usiug  the  determinant  notation  (§  36),  we  get 

(1) 


A  =1 


X, 

yx 

1 

x^ 

y. 

1 

X, 

y> 

1 

I 


For  example,  the  area  of  the  triangle  with  the  vertices  (0,  2), 
(3,  0),  (4,  1)  is 


0     2 

3  0 

4  1 


151.     Theorem.      The  graph  of  an  equation  of  the  first  degree 

(2)  Ax -\- By +0=^0 
is  ahuays  a  straight  line. 

Consider  three  points   {x^,  y^),  {x^,  yj,  (x^,  y^  of  the  graph. 
Then 

(3)  Ax,-^By,+  C=^0,  Ax,  + By,-^  C  =Q,  Ax,  + By,+ C  =^0. 

The  area  of  the  triangle  whose  vertices  are  the  three  points  is 
given  by  formula  (1).  li  A  4^  Q,  we  multiply  the  elements  of  the 
first  column  of  the  determinant  by  Ay  the  elements  of  the  second 
column  by  B^  the  elements  of  the  third  column  by  (7,  and  add  the 
last  two  sets  of  products  to  the  first  set.     We  have  (§§  45,  47) 


Ax,^By,JrG    y,     1 

0    y,    1 

^  A  =i 

Ax,^By,+  C    y,     1 
Ax,  +  By,  +  C    y,     1 

=  1 

0    y,    1 
0   y,   1 

=  0, 

upon  applying 

the  hypotheses    (3).      E 

^ence 

A   =  0,  S< 

)  that  the 

three  points  lie  in  a  straight  line.  Let  the  third  point  {x^ ,  y^  move 
to  any  new  position  on  the  graph.  As  the  area  is  again  zero,  the 
new  point  lies  on  the  straight  line  determined  hy  {x^,  y^)  and 
(^2 '  ^2)-  Hence  every  point  of  the  graph  lies  on  that  straight 
line. 

If  ^  =:  0  and  B  ^  Oy  we  multiply  the  elements  of  the  second 
[column  by  B  and  add  to  them  the  products  of  the  elements  of  the 


15S  GRAPHIC  ALGEBRA;    SIMULTANEOUS  EQUATIONS.  [Ch.  XVIII 

third  column  by  C,  The  resulting  determinant  has  three  zeros  in 
the  second  column,  so  that  ^  A  =  0,  whence  A  =  0.  In  this  case 
the  line  (2)  becomes  By  -\-  C  =  0  and  is  parallel  to  the  a;-axis,  since 
the  ^-coordinate  remains  constant. 

It  remains  to  prove  that,  inversely,  every  point  on  the  straight 
line  joining  Qx^^  y^)  and  {x^,  y^)  lies  on  the  graph  of  (2).  This 
result  is  shown  in  §  153  below. 

152.    To  find  the  coordinates  of  the  point  which  divides  in  a 
given  ratio  the  straight  line  joining  tivo  given  poiiits. 

Let  the  first  point  P^  be  {x^ ,  y^  and  the  second  point  P^  be 


p. 

%-y< 

p,/ 



'          

Fig.  6. 


{x^,  y.^).  Let  the  given  ratio  be  m^-.m^.  Let  P,-  be  the  point  which 
divides  P^P^  internally  in  this  ratio,  and  P^  be  the  point  which 
divides  PiP^  externally  in  this  ratio.  Let  their  coordinates  be 
{Xi,  y,)  and  (cr,,  y^,  respectively. 

Since  P^Pi\  P^Pi  -■  r)i^'-  ^^2  ^^^  ^i^e '  P^Pe  —  '^^I'l^h^  we  have 


-  Vx  _  ^i  —  ^1  _  ^^^ 

-    Vi            ^2    -   ^i            ^^'2' 
^     1     /       '              z      1 

ih  -Ve  _^i-X, 

_  »«-. 

Hence 
(4) 

(5) 

y%  -  Ve     •-»-,  -  X, 

■''         «/,  +  m,    ' 

«, 

«»i  +  m,    ' 

^^         m,  —  m,    ■ 

Sec.  153]  COLLEGE  ALGEBRA.  159 

Corollary.       The    middle   point   of    the  straight   line  wining 
{x^ ,  ?/J  and  {x^ ,  y,)  is 

(6)  (^%  H^). 

Note.     The  dista^ice  hetween  tivo p)oints  {x^,  y^  and  (x^,  y^  is 


(7) 


V{^,  -  x^f  +  (y,  -  yJK 


P. 


R 

y^-y. 

Pe 

X    ^i-^« 

1 
1 
1 
1 

i 

Fig. 


153.  We  may  now  prove  that  if  the  graph  of  Ax  -\-  By  -\-  C  =0 
contains  two  distinct  points  (2-^,  y^  and  {x^,  y^)^  it  contains  every 
point  on  the  straight  line  joining  them.  By  hypothesis,  the  first 
and  second  relations  (3)  are  satisfied.  In  view  of  the  last  section,  it 
remains  to  be  proved  that  the  points  P^and  P^  with  the  coordinates 
(4)  and  (5),  respectively,  lie  on  the  graph  of  Ax  +  By  +(7=0, 
whatever  be  the  valnes  of  m^  and  m^.     The  condition  for  F^  is  that 


Ax,+  By,+  C= 


-^{Ax^+  By,)  +  -^ 


{Ax,+  By,)  +  0 


shall  vanish.    In  view  of  the  first  and  second  relations  (3),  it  equals 


-(-  0)  + 


Wj  +  7n,^  m,  +  m, 

Similarly,  Ax^  +  By^  +  O  vanishes. 


{-G)  + 


l6o  GRAPHIC  ALGEBRA;    SIMULTANEOUS  EQUATIONS,  [Ch.  XVIII 


I 


164.  From  what  precedes,  every  straight  line  is  the  graph  of 
some  equation  of  the  first  degree.  If  two  points  (x^,  y^  and  {x^,  y^ 
on  the  line  are  given,  the  required  equation  may  be  written 

X     y      \ 
(8)  x^    y,  A    =0. 

Indeed,  upon  expansion  according  to  the  elements  of  the  first  row 
(§46),  there  results  an  equation  of  the  first  degree  in  a;. and  y, 
which  must  therefore  represent  some  straight  line  (§  151).  But  the 
line  coincides  with  the  given  straight  line.  In  fact,  it  contains  the 
point  {x^y  ^j),  since  the  replacement  of  x  by  x^^  and  y  by  y^  in  the 
determinant  in  equation  (8)  gives  a  determinant  having  the  first  and 
second  rows  alike,  so  that  the  equation  is  satisfied  (§  46).  Similarly, 
it  contains  the  point  {x^,  y^), 

A  second  proof  follows  from  §  150,  since  the  area  of  the  triangle 
formed  by  {x,  y),  {x^,  y^),  {x^,  i/J,  is  one-half  the  determinant  in 
(8);  but  the  area  must  be  zero  if  the  point  {x,  y)  lies  on  the  line 
joining  {x^,  y,)  and  {x^,  y^). 

155.  Example  1.  Find  the  equation  of  the  straight  line  making  an  inter- 
cept a  on  the  a^-axis  and  an  intercept  b  on  the  ly-axis. 

The  line  through  the  points  {a,  0)  and  (0,  b)  has  the  equation 

=  ab  —  ay  —  bx  =  0, 


=  1. 


X 

y 

1 

a 

0 

1 

0 

b 

1 

It  may  be  written  in  the  convenient  form  — \-  ^ 

''  a       b 


Example  2.  The  medians  of  any  triangle  meet  in  a  point  and  each  median 
is  trisected  by  that  point. 

Let  the  vertices  Pi,  Pg*  ^3 1^^  (^i»  2^1  )>  (^2»  ^2)*  (^3»  ^s^  respectively.     By  the 

I  -h52     Vi  4-  y2\ 
2     '         2      / 
Let  Q  be  the  point  which  divides  the  sect  P^M^^  internally  in  the  ratio  2  :  1. 
By  formula  (4),  the  coordinates  of  Q  are 


corollary  of  §  152,  the  middle  point  M^^  of  the  side  P^P^  isf — 

which  divides 
coordinates  of 


3!i  +  m^  +  Xt 


i  +  1 


Sec.  155]  COLLEGE  ALGEBRA,  i6l 

^*-  2  +  1  ~  3 

Similarly,  the  middle  point  My^  of  the  side  P^P^  is  ^^-i^^     ^^^r  ^^^ 

the  coordinates  of  the  point  which  divides  the  sect  Pg^is  internally  in  the 
ratio  2  :  1  are  seen  to  be  the  values  aj/,  yt  just  given.  In  a  similar  manner,  or 
by  the  symmetry  of  the  result,  the  point  [xi,  yi)  divides  the  sect  P^M^^  inter- 
nally in  the  ratio  2  :  1,  ifjs  being  the  middle  point  of  the  side  PjA-  Hence 
the  medians  meet  in  a  point  and  trisect  each  other. 

Example  3.  Find  the  equations  to  the  straight  lines  which  pass  through 
the  point  (1,  —  3)  and  make  equal  intercepts  on  the  two  axes. 

By  Ex.  1,  the  required  lines  have  the  equations 

a       —  a  —b       —  b 

since  a  line  through  (1,  —  3)  will  either  have  a  positive  intercept  on  the  aj-axis 
and  a  negative  intercept  on  the  y-axis  or  else  have  negative  intercepts  on  both 
axes  (as  a  diagram  will  show).  Substituting  1  for  x  and  —  3  for  y,  we  have 
for  the  respective  lines  the  conditions 

\  a^--a'-b^-b 

whence  a  =  4,  b  =  2.     The  lines  are  x  —  y  =  4:  and  x  -{■  y  =  —  2. 

Example  4.  To  find  the  coordinates  of  the  center  of  the  circle  inscribed 
in  a  given  triangle. 

Let  the  vertices  P^,  Pg,  Pg  be  (x^,  2/i)»  (^2»  ,V2^>  (^3*  ^s)  ^^d  the  opposite  sides 
«!,  s,^,  S3.  The  bisector  of  the  interior  angle  at  P^  meets  the  side  P2P3  at  a  point 
§1  which  divides  it  internally  in  the  ratio  s^  :  8.2.  Hence  the  co5rdinates  of  Q^ 
are  (§  152) 

«3  -{-  ^2  *'3  "4"  ^2 

But  P2Q1 :  P3C1  =  «3  :  «2  gives  P^Q^  +  P.^Q^  :  P^Q^  =  s^  +  8^  :  s^,  whence 

The  bisector  of  the  interior  angle  at  P3  meets  PiQi  at  the  required  center  (7/ 
Hence  CMivides  P^Q^  internally  in  the  ratio  s^  :  P^Qi,  viz.,  Sj  •  o"-  Hence  the 
coordinates  of  G  are 


yc  — 


/^3^3   ~r   ^2^2\ 

8^X^  +  S^X^  +  8^X^  ^   gl.Vl  +  ^2^2  +  ^3^3 

«1  +  «2  +  «3  ""  «1  H-  «2  4-  «3       ' 


i62   GRAPHIC  ALGEBRA ;    SIMULTANEOUS  EQUATIONS.  [Cii.  XVIII 

Thus,  if  the  vertices  are  (0,  12),  (5,  0),  (35,  0),  we  get  (Note,  §  152)  s^  =  30^ 
«2  —  37,  Sg  =  13.     The  coordinates  of  (7  are  therefore  Xc  =  8,  Pc  =  4^. 

EXERCISES. 

Find  the  equation  of  the  straight  line 

1.  Cutting  off  3  on  the  ic-axis  and  4  on  the  y-axis. 

2.  Cutting  off  —  2  on  the  a;-axis  and  —  5  on  the  ^-axis. 

3.  Passing  through  the  points  (0,  0)  and  (3,  —  3). 

4.  Passing  through  the  points  (5,  6)  and  (3,  4). 

5.  Passing  tlirongh  the  point  (1,  4)  and  cutting  off  —  3  on  the  ?/-axis. 

6.  Find  the  equations  of  the  diagonals  of  the  rectangle  formed  by  the 
lines  X  =  2,  a;  —  —  3,  2/  -  4,  ?/  =  —  5. 

7.  Find  the  equations  of  the  lines  through  the  origin  and  the  points  trisect- 
ing the  segment  of  the  line  i^x  -\-  ^  =  12  intercepted  between  the  axes. 

8.  Show  that  the  coordinates  of  the  centers  (Jj,  O2,  C\  of  the  circles  escribed 
to  the  triangle  with  the  vertices  {x-^y  y^),  {x^,  y^,  (ojg,  2/3)  and  the  opposite  sides 

*1>  *2»   *3>   ^^® 

~  ^1^1  +  ^2^2  +  ^3^3^        -  ^l.Vl  +  ^2^2  -V  hVz         (f(^i.   Q^  oppOSitC  S^\ 
■^  «!   +  Sj  -j-  %  ~  *1  +  ^2  ~i~  **3 

s^x^-8.,X2-^s^x^^     s^y^  -  ^2^2  +  ^3.^3      (for  G,  opposite  s.), 

*i         *2    r  *3  ^1  —  ^2  +  *3 

with  similar  expressions  for  Cg,  the  sign  of  s^  being  changed. 

9.  Find  the  centers  of  the  inscribed  and  escribed  circles  for  the  triaui^b 
whose  vertices  are  (9,  0),  (0,  12)  and  lO,  40). 

10.  Find  the  centers  of  the  inscribed  and  escribed  circles  for  the  triangle 
w^hose  sides  have  the  equations  4a?  —  3y  =  0,  ^x  —  Ay  4-  12  =  0,  dx-\-4:y-^2 --  U. 

11.  Find  the  equations  of  the  sides  of  the  triangle  with  the  vertices 
(-1,  -2),  (2,  -3),  (1,  4). 

Find  the  coordinates  of  the  points  which  divide  internally  and  externally 

12.  the  line  joining  (1,  2)  and  (4,  5)  in  the  ratio  2:3. 

13.  the  line  joining  (—  2,  —  3)  and  (-  4,  7)  in  the  ratio  5 :  6. 

14.  the  line  joining  (0,  0)  and  (6,  8)  in  the  ratio  4  : 5. 

15.  the  line  joining  {c  -}-  a,  c  —  d)  and  (c  —  d,  c  -]-  d)  in  the  ratio  c  :  d. 

16.  The  lines  joining  the  middle  points  of  opposite  sides  of  a  quadrilateral 
and  the  line  joining  the  middle  points  of  its  diagonals  meet  in  a  point  and 
bisect  one  another. 

17.  If  Pi ,  F-i,  ...,  Pn  denote  the  points  (a?, ,  yj),  (X2 ,  ^2),  .  .  ,  ,  (x„ ,  y„), 
and  P1P2  is  bisected  at  the  point  M\;  M^P-i  is  divided  at  M-^  in  the  ratio  i  :  2  ; 
M^Pa  is  divided  at  Mz  in  the  ratio  1:3;  /I/3A  is  divided  at  M^  in  the  ratio 
1  : 4;  etc.;  prove  that  the  final  point  Mn-i  has  the  coordinates 

1  1 

-{xx  +  aJa  +  aJa  +  .  .  .  +  Xn\         ~{yx  +  2^2  +  2/3  +  •  •  •  +  Vn). 


Sec.  156] 


COLLEGE  ALGEBRA, 


163 


18.  Show  that  the  area  of  the  polygon  P1P2  .  .  .  Pn  of  Ex.  17  is 


19.  Find  the  area  of  the  polygon  whose  vertices  are  (4,  —  7),  (5,  —  2), 
(3,4),  (1,1). 

20.  Find  the  area  of  the  triangle  with  the  vertices  {a,  b),  {—  a,  b  -\-  c), 
{—  a,  c  ~  b), 

21.  Find  the  coordinates  of  the  point  of  intersection  of  the  two  lines  join- 
ing the  middle  points  of  the  opposite  sides  of  the  quadrilateral  whose  vertices, 
taken  in  order,  are  (0,  3),  (0,  0),  (4,  0),  and  (6,  7). 

156.  To  find  the  graph  of  tlie  equation  y^  =  4:X,  we  observe  that 
a  negative  value  of  x  gives  an  imaginary  value  of  y,  so  that  the 
graph  lies  to  the  right  of  the  y-axis.  For  each  positive  value  of  x 
til  ere  are  two  values  of  y,  namely  2\/x  and  —  2\/x,  so  that  the 
graph  is  symmetrical  with  respect  to  the  i?;-axis.  Plotting  the 
pairs  of  points  (1,  +  2),  (1,  -  2),  (2,  24/2),  (2,  -  24/2),  (4,  4), 
(4,  -  4),  (16,  8),  (16,  -  8),  ...  ,  and  the  point  (0,  0),  we  may 
trace  the  graph  AOB  in  Fig.  8.     The  graph  is  called  a  parabola. 


Y 

A 

^ 

■^ 

^ 

y^ 

,^ 

^-• 

^ 

/ 

^ 

^ 

\ 

/ 

^ 

\ 

(/ 

\ 

0 

( 

X 

X 

\ 

1 

\ 

X 

/ 

\ 

\ 

N 

/ 

^ 

^ 

^ 

k. 

^ 

y 

■ 

"^ 

^ 

^ 

Fig.  8. 


The  figure  also  gives  the  graph  of  the  equation  x^  -\~  y^  =  10:r, 
which  may  be  written  {x  —  5)^  -\-y^  ~  25.  Since  the  latter  is  derived 
from  X^  -(-  ^2  __  25^  the  circle  of  Fig.  4,  by  writing  a;  —  5  in  place 
of  X,  it  is  likewise  a  circle  of  radius  5,  but  referred  to  a  set  of  axes 


1 64  GRAPHIC.  ALGEBRA;    SIMULTANEOUS  EQUATIONS.  [Ch.  XVIII 

of  coordinates  obtained  by  moving  the  previous  ?/-axis  5  units  to 
the  left.  Hence  the  graph  of  ^c^  +  ^^  _  ^^q^  |g  ^  circle  of  ladius  5 
and  center  (5,  0).  Its  intersections  with  the  parabola  y'^  =  4:X  are 
seen  to  be  the  three  points  (0,  0),  (6,  |/24),  (6,  —  4/24),  found  by 
solving  the  equations  considered  to  be  simultaneous. 

Similarly,  y^  =  4:X  intersects  the  circle  x^  -{-  y^  =  25  in  just  two 
points;  it  intersects  the  circle  {x  —  6)*  -\- y^  =  25,  with  radius  5 
and  center  (6,  0),  in  four  points;  it  intersects  the  circles 
{x  —  c)'^  -{-y'^  =  25,  of  radius  5  and  center  (c,  0),  in  no  point  if 
c  >  7^.  In  fact,  if  we  eliminate  y^  from  the  latter  equation  by 
means  of  y^  =  4:X,  we  get  a  quadratic  for  x,  with  the  roots 


6'  -  2  ±  |/29  -  4c. 

(x  +  5)2  +^2  =  25  and  y^  —  4x  intersect  only  at  the  origin  (0,  0). 

Two  simultaneous  equations  of  the  second  degree  ifi  x  and  y  may 
have  4,  e3,  2,  1,  or  no  sets  of  real  solutions  x,  y. 

157.  In  order  that  the  three  simultaneous  equations 

(9)     a^x  +  h,y  -\-c^  =  0,      a^x  +  h^y  -\-  c^  =  0,      a^x  +  %  -\-c^  =  0 
shall  have  a  common  set  of  solutions  x,  y,  the  determinant 


n- 


^1   \   ^1 

Cf,      \      c, 
(7,       Z>,       C^ 


must  vanish.  This  follows  from  §  44  when  we  set  2;  =  1.  To  give 
another  proof,  multiply  the  elements  of  the  first  column  by  x  and 
the  elements  of  the  second  column  by  y  and  add  the  results  to  the 
corresponding  elements  of  the  third  column;  the  new  elements  of 
the  third  column  will  then  be  zero,  if  x  and  y  be  the  common  solu- 
tions.    Hence  must  D  =  0. 

Inversely,  if  B  =  0,  equations  (9)  have  at  least  one  set  of  solu- 
tions X,  y  in  common  (§  44).  In  fact,  the  third  equation  (9)  then 
follows   from  the  first  and  second   equations.     Their  graphs  are 


Sec.  158] 


COLLEGE  ALGEBRA, 


165 


straight  lines  which,  in  general,  intersect  in  a  point  (ic,  y).     But 

the  second  and  third  equations  both  follow 

from  the  first  if  all  the  minors  of  D  vanish 

(§  44),  in  which  case  there  are  innumerable 

sets  of  solutions  x,  y  in  common.     The  three 

graphs  are  then  coincident  lines. 


For  example,  the  lines  (Fig.  9) 


RP 
OP 
SP 


2x  +  y  -  10  =  0, 
4:X  —  3y  =z  0, 

2x^y  -    2  =  0 


The 


\ 

Y 

n 

■\ 

/ 

/ 

R 

\, 

/ 

/ 

/ 

\ 

/ 

' 

/ 

\ 

/ 

/ 

\ 

y 

\ 

y 

\ 

n 

y 

\^ 

y 

\ 

/ 

\ 

0 

/ 

\ 

X 

/ 

/ 

\ 

's 

/ 

\ 

/ 

• 

k 

/ 

\ 

Fig.  9. 


intersect  in  the  point  P,  viz.,  (3,  4). 
determinant  D  is  seen  to  be  zero. 

158.   Theorem.       Two   points    {x^y^    and 
(^2^/2)  ^^^   ^^  ^^^^  same  side  or   on   o^)posite  sides  of  the  graph  of 
Ax  -\-  By  -\-  O  =:  0  cxcording  as  the  quantities 

Q.^Ax,  +  By,  +  C,         Q,=  Ax,  +  By^+ 0  _ 

are  of  the  same  sign  or  of  opposite  signs. 

Through  (x^ ,  y^)  draw  a  parallel  to  the  ^-axis  and  let  it  meet 
the  graph  of  the  given  line  in  the  point  {x^ ,  ?//)  so  that, 

Ax,  +  By^'+  C  =  0,     y,'=-{Ax,+  C)^B. 

'-'     y,-y^^U^^.+  By,+  C)^B^QJB.     , 

Through  {x^.  y^  draw  a  parallel  to  the  «/-axis  and  let  it  meet 
:  the  given  line  in  the  point  {x^^  y^).     As  before,  y^  —  yl  —  QJB, 

Then  (x^,  y^  and  {x^,  y^  are  on  the  same  side  or  on  opposite 
sides  of  the  given  line  according  as  y^  —  y^  and  y^  —  y^  are  of  the 
same  sign  or  of  opposite  signs,  i.e.,  according  as  Q^  and  Q^  are  of 
the  same  sign  or  of  opposite  signs. 

Corollary.     The  point  [x^,  y^  and  the  origin  ^0,  0)  are  on  the 


1^6       GRAPHIC  ALG.;    SIMULTAhlBOUS  INEQUALITIES.    [Cn.  XVITI 

same    side   of   the   graph   of   Ax  -\-  By  -\-  C  —0   if,  and  only  if, 
Ax^-^-  By^  +  C  ^^^  ^  \\^YQi  the  same  sign. 

159.  A  point  {x^,  y^)  lies  inside,  on,  or  outside  a  circle  of  radius 
r  and  center  at  the  origin  (0,  0)  according  as 

x^^  +  ^1^  —  ^'''^  <  0,     =  0,     or     >  0. 

In  fact,  the  equation  of  the  circle  is  x^  -\-  y^  —  r^  (compare  §  147). 
A  point  {x^,  y^)  lies  inside,  on,  or  outside  the  graph  of 

fl-i-  ^    -  1 
a'  ~^  d-'    ~ 

X  ^  -9/2 

(called  an  ellipse)  according  as  ~  -\-  ~  —  1  <  0,     =0,    or    >  0, 

€1/  0 

In  proof,  draw  through  the  point  {x^,  y^  ar  parallel  to  the  ^-axis 
and  let  it  meet  the  graph  in  a  point  {x^^  y/),  so  that 

£l   f   li^  -  1      • 

For  the  parabola  ^^  =  \x  whose  graph  AOB  is  given  in  Fig.  8, 
we  will  say  that  a  point  lies  within  the  parabola  when  it  lies  on  the 
same  side  of  the  graph  as  the  line  OX,  By  a  proof  similar  to  that 
just  employed,  we  find  that  the  point  (x^^  y^  lies  within,  on,  or 
without  the  graph  of  y'^  =  4:X  according  as  y^^  —  4x^  is  less  than, 
equal  to,  or  greater  than  zero. 

160.  The  preceding  results  enable  us  to  treat  .very  simply  prob- 
lems in  simultaneous  inequalities. 

Example  1.  Resolve  the  simultaneous  inequalities 
fi~x-y+l>  0,    f,  =  2x-{-y-6>0,    /3  =  -  5^  +  2y  +  10  >  0. 

The  graphs  of  /^  =  0,  f  ^  =  0,  f^  =  0  are  given  in  the  accompanying 
figure.     By  the  corollary  of  §  158,  f^  is  positive  for  all  points  on  that  side  of 


r 


8ec.  160] 


COLLEGE  ALGEBRA. 


the  graph  of /^  =  0  on  which  the  origin  lies, 
tive  only  for  the  points  on  the  opposite  side 
three  inequalities  are  therefore  satisfied 
simultaneously  by  all  the  points  in  the 
shaded  triangle  and  by  no  other  points. 
The  vertices  of  this  triangle  are  (f,  |), 
(4,  5),  p/,  Jg").  Hence  the  solutions 
X,  y  are  such  that 


Similarly  for/g;  but /a  is 
of  /2  =  0  from  the  origin. 


107 

posi- 
The 


<  ic  <  4, 


<y  <^. 


Example  2.  Resolve  the  simultane- 
ous inequalities 
2^2  _  4^  >  0,     x^  -  10a'  4-  y2  ^  0. 

The  sets  of  values  x,  y  must  be  such 
that  the  corresponding  points  {x,  y)  lie 
outside  of  the  parabola  y^  —  ix  and 
inside  of  the  circle  {x  —  5)'^  -|-  y'^  =  25. 
These  points  therefore  lie  within  the 
two  crescent-shaped  areas  in  the  figure 
of  §  156.     Hence  the  solutions  are 

V2i  <y  <  V^, 
>  Ax, 


,       0  <x  < 


\ 

/ 

/ 

Y 

\ 

/ 

\ 

^ 

f 

\ 

^ 

\^ 

k4'A 

f 

__ 

A 

Si_ 

7 

^ 

=u 

M^ 

— 

— 

=^i=i 

/ 

^=\\ 

W 

/ 

"^ 

1 

/ 

/ 

\ 

/ 

\ 

0 

/ 

\ 

X 

/ 

\ 

/ 

\^ 

\ 

\ 

1 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

L 

\ 

r 


Fig.  la 


EXERCISES. 


1 

3.  x' 

5 

6 


Plot  the  graphs  and  find  the  common  solutions  of 
x^  ^  y-^  =  25,     y^  =  4x-\-  24.      2.  x'  -j-  f  =  25, 
x-^ 


r 


25, 


86^   9 


dx -j- Sy  -{-  5  =:  0, 

X 


a'^  b 


1, 


b  ^  a 


1.        4.  a;2  +  y^  =z  16, 
3^  4-  4^  -f  6  =  0,     Qx  -\-  5y  - 
1,     y  —  X  =  0. 


7. 


p' 

=    4:X   + 

28. 

x' 

^4y'=. 

64. 

-  9 

rr   0. 

4^-2 

+  V  = 

■  36. 

a?  --  2^  -  3  ==  0,     4y  +  6  -  2x.     8.  x"^  -{-  y"^  = 
Solve  graphically  the  simultaneous  inequalities 
9.  X  -  y  >  0,     -3x  -\-  2y  i-  6  >  0,     ic  +  2^  -  4  >  0. 

10.  X  -  y  -]-  4:  >  0,     X  -  2y  -\-  5  <  0,     a-  +  5y  -  4  <  0. 

11.  X  +  y  -  d  >  0,     2a;  -  3y  +  1  >  0,     2a'  -  3^  -  5  <  0. 

12.  x^  +  2^2  -  25  <  0,     2a  H-  2/  -  10  >  0. 
i;;.  a2  H-  2/'  -  25  <  0,     2a  +  2^  -  10  <  0,     a  +  2^  -  2  >  0. 

14.  y^  <  4a,     {x  -  6f  -f  y'  <  25,     y  >  0. 

15.  The    equation    to    any    straight    line    through    the    intersection    of 
\ax  -{-  by  -\-  c  =  0   and  Ax  -{-  By  -\-  C  =  0  may  be  written 

(ax  -\-  by  +  c)  i-  m{Ax  -\-  By  -{-  C)  =  0. 


1 68       GRAPHIC  ALGEBRA;    NUMERICAL   EQUATIONS.      [Ch.  XVIII 


16.  Find  the  eqaation  of  the  straight  line  th»")ugh  (0,  1)  and  the  inter- 
section oi  y  -\-  X  =  0  with  x  —  'dy  -[-  1  =  0. 

SOLUTION  OF  NUMERICAL  EQUATIONS. 
161.  Consider  first  the  quadratic  equation  i?;^  —  6a;  —  3  =  0.  The 
graph  of  y  =  o;'^  —  6a;  —  3  is  given  in  the  figure.  It  is  roughly  deter- 
mined by  the  points  (  —  1,  4),  (0,  —  3), 
(1,-8),  (2,  -11),  (3, -12),  (4,-11), 
(5,  -  8),  (6,  -  8),  (7,  4).  The  sym- 
metry  with  respeot  to  the  line  x  —  ^  fol- 
lows from  the  form  y  -\-  12  =  (x  —  2y 
in  which  the  equation  may  be  written. 
The  dotted  lineF  "have  the  equations 


l_Y           i 

r    J 

1X4 

\  0          !                  X 

4      it      1 

4^  m   ± 

T     t 

r    ^         L 

V    J - 

X   ^  2 

4  iL  i 

^It  t 

-       X-^4 

4^4 

)  '  / 

IV|/ 

y'~y  +  y^ 


0, 


X'  ^x  • 


3  =  0. 


Keferred  to  them  as  new  coordinate  axes, 
the  equation  is  y'  =  x''^.  To  this  form 
any  equation  of  the  form  y  =  x^-{-  px  +  q 
may  be  reduced  The  graph  is  called 
Fi<*-  "•  a    parabola.*      To   solve    the    equation 

^2_g^_3  _  Q  consists  in  finding  the  values  of  x  which  makey  =  0. 
In  the  graph,  it  consists  in  finding  the  a:-coordinates  of  the  points 
in  which  the  graph  crosses  the  a;-axis.  The  one  lies  between  0  and 
—  1,  and  the  other  between  6  and  7;  the  exact  values  are  evidently 
3  ±  |/l2.  It  is  clear  that  all  quadratic  equations  in  which  tlie 
coefficient  of  x^  has  been  made  unity,  lead  to  the  same  graph,  the 
variation  being  the  relative  position  of  the  axes  of  coordinates. 

162.  Consider  the  cubic  equation  x^  +  ^x^  —  7=0.  The 
graph  of  y  =z  x^  -\-  ^x^  —  7  is  given  in  the  figure,  as  sketclied 
through  the  points  (-  4,  -  7),  (-  3^,  -  -J),  (-  3,  2),  (-  2^,  2|), 
(-2,1),     (-1|,  -1|),     (-1,-4),     (-^,-61),     (0,-7), 

*  The  parabola  y'^  =  4a;,  whose  graph  is  given  in  §  156,  may  be  reduced  to 
the  above  form  by  taking  x'  —  y,  y'  —  4x.  This  transformation  turns  the 
picture  through  90"  and  magnifies  the  resulting  figure  in  the  vertical  direction. 


Sec.  163] 


COLLEGE  ALGEBRA, 


169 


Y 

r~\ 

'     \ 

i 

/ 

0 

X 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

{h  -  H)>  (1.  -  2).  (1-1.  -  -829),  (1.2,  .488),  (1.25,  1.203), 
(1.5,  5|).  The  roots  of  the  cubic  lie  between  1  and  2,  between 
—  1  and  —  2,  and  between  —  3  and  —  4. 
More  closely,  the  positive  root  lies  be- 
tween 1.1  and  1.2.  To  determine  the 
second  decimal  place,  the  process  of 
direct  trial  of  1.11,  ...,  1.19  becomes 
laborious.  To  shorten  the  work,  set  x 
=  1.1  -{- z,  whence  the  relation  becomes 

(10)  y  =^  z^  -\~  7.3^2  _|_  12.43^  -  .829. 

In  determining  z  to  make  y  =  0,  the 
terms  z^  and  7.3z^  are  of  little  conse- 
quence, since  z  is  of  the  denomination 
hundredths.  We  should  expect,  there- 
fore, that  z  lies  between  .06  and  .07,  being  given  approximately 
by  12 A3z  —  .829  ~  0.  In  fact,  y  is  negative  when  z  =  .06  and  is 
positive  when  z  =  .07.     Setting  z  =  .06  -|-  w,  the  relation  becomes 

(11)  y  =  tv^+  7ASw^  +  13.3168W  -  .056704. 

Using  the  last  two  terms  only,  we  find  that  ?/  =  0  if  w  =  .0042  +, 
to  four  decimal  places.  In  verification,  to  =  .0042  makes  y  nega- 
tive and  w  =  .0043  makes  y  positive.  Hence  the  positive  root  of 
the  cubic  is  1.1642  +. 

163.  The  calculations  may  be  shortened  by  Horner's  method,* 
which  is  based  upon  the  process  called  Synthetic  Division. 

The  process  of  the  division  of  x^  -f  4:x'^  —  7  by  a;  +  3  may  be 
exhibited  as  follows: 


Fig.  12. 


x^  +  3x^ 

X^ 

—   '7 

X  -{-  3           =  divisor 

x^  -\-  X  —  3  =  quotient 

X^  +  Zx 

-  3x 

-  3x 

-  9 

+  2  =  remainder. 

*  Published  in  1819  by  W.  Gt.  Horner. 


I70       GRAPHIC  ALGEBRA;    NUMERICAL   EQUATIONS,       [Ch.  XVIII 

Since  the  literal  terms  x^,  x^^  x  may  be  supplied  from  their  relative 
positions,  we  may  indicate  the  work  by  detached  coeflicients : 


14     0 
1     3 

-  7 

1 
1     3 

-3 
~  3 

-  9 

1     3 


divisor 


LI     —  3  quotient 


2     remainder. 

We  agree  to  omit  the  first  term  of  each  partial  product  (1,  1,  —  3, 
respectively)  since  it  is  merely  a  repetition  of  the  number  above  it. 
Next,  if  the  second  term  of  the  divisor  be  changed  in  sign,  the  new 
divisor  being  1  —  3,  the  sign  of  the  second  term  of  each  partial 
product  will  be  changed,  so  that  the  subtraction  previously  per- 
formed is  replaced  by  addition.  The  work  for  the  division  by 
a;  -}-  3  thus  becomes  the  following : 


1-3 


+  9 

2 


Arranging  the  work  in  a  horizontal  row  and  omitting  the  coefficient 
1  in  the  divisor   1     —  3,    the  work  stands  thus: 


Dividend  =1  4  0-7 

Partial  products  =         —3—3+9 

Quotient  =1  1—3",        2^ 


-  3 


remainder. 


In  this  final  form  the  process  is  called  synthetic  division. 
We  next  divide  3a;'  +•  x^  —  ^Ix^  +21^  _  5  by  a;  —  2. 

Dividend  ==3     1       0-31  0  0  21-5 

Products  ^        6     14         28     -  G     -  12     -  24     -     6 


+ 


Quotient   =  3     7     14      -  3     -  6 


12     -     3, 


II: 


;  rem. 


Sec.  164]  COLLEGE  ALGEBRA.  171 

# 

The  quotient  is  dx^  +  Ix^  +  14.t*  _  3^^  ~  6^2  _  i^x  -  3,  and 
the  remainder  is  —  11. 

164.  Ee turning  to  the  example  and  graph  of  §  162,  we  see  that 
one  root  is  a;  ==  1  +  ^^,  n  being  a  positive  decimal.     Then 

y={\  +  uf  +  4(1  +  uf  _  7  =  ?^3  +  72^2  +  11?^  -  2. 

Without  the  expansion  just  made,  the  final  form  ii^ -{-  ,  ,  .—2 
may  be  obtained  by  synthetic  division.  If  the  degree  of  the  given 
equation  is  large,  say  6  or  7,  tlie  second  process  is  much  shorter 
than  the  first.     Since  ii  =  x  —  \^  we  have 

^  =  2;3  +  4a;2  -  7;     y=  (x  -  1)^  +  7{x  -  If  +  ll(a^  _  1)  _  2. 

Hence  if  we  divide  the  first  expression  by  x  —  1,  the  remainder 
must  be  —  2,  as  is  the  case  with  the  second  expression.  The 
respective  quotients  are  x^  -{-  dx  -{-5  and  {x  —  1)^  -|-  7{x  —  1)  -f-  H? 
and  they  must  be  equal.  Dividing  them  by  iz:  —  1,  the  quotients 
are  x  -}-  6  and  {x  —  1)  +  ^y  ^ii^  the  remainders  are  each  11.  It  is 
thus  clear  that  in  the  division  of  x^  +  4^^  —  7  by  :c  —  1  the  suc- 
cessive remainders  —  2,  +  H^  '^»  1  must  be  coefficients  in  reverse 
order  of  the  transformed  equation  tt^  +  '^^^^  +  H^^  —  2-  ^Y  syn- 
thetic division  this  work  may  be  done  thus: 

1        4        0-7    I  1 
15  5 


5        5,    —  2  =  first  remainder 
1        6 


1        6,    11  =  second  remainder 

!_ 

1  ,     7  =  third  remainder. 

The  general  theorem  involved  is  stated  in  the  next  section. 

165.  Giveny  =  p^x""  -{-p^x''-'^-^- p^x''-^^  +  •  .  •  +Pn-i^+Pn  «'^^ 
X  =^  u  -\-  Ji,  to  oMain  the  expression  for  y  in  terms  of  u,  loe  divide 
p^x''  -\-  '-'-{-  Pn  ^y  ^  —  h  and  call  the  quotient  Q^  and  the  remainder 


172       GRAPHIC  ALGEBRA;    NUMERICAL  EQUATIONS,       [Ch.  XVIII 

rp  then  divide  Q^hy  x  —  h  and  call  the  quotient  Q^  and  the  re- 
mainder r^,  etc.     The  reqm^^d  expression  for  y  is 

Po"^""  +  ^n^f^''~'  +  .  .  .  +  r,u^  +  r,u  +  r^. 
In  proof,  we  observe  that  the  final  expression  equals 
F^  plx  -  hf  +  rr,{x  -  70— ^  +  .  . .  +  r,{x  -  hf  +  r,{x--Ji)  +  r,. 

To  prove  that  E  is  another  form  of  the  initial  function 
y  z=z  p^x"^  -\- ,  , ,  -\-  p^^  we  divide  both  E  and  y  hy  x  —  h.  By 
inspection,  the  quotient  ol  E  is 

p,{x  -«  hf-'  +  r^[x  -  hy-^  +.  .  ,+  r,{x  -  h)  +  r, 

and  the  remainder  r^  By  the  notations  of  the  theorem,  the  divi- 
sion of  y  =  PqX^  -{-...  -\-  pnhj  X  ~  h  yields  the  quotient  Q^  and  the 
remainder  r^     Hence  will  E  =  y  if 

Qi  =Po{^  -  ^'T-'  +  rn{x  -  hy-'  +  .  . .  +  r,{x  -  A)  +  r,. 
Dividing  each  member  hjx  —  h^  the  quotients  are  respectively 
(12)  (?„    Po{^-'^Y~'  +  rn{x-hf-'+,..  +  r, 

and  the  remainder  is  r^  in  each  case.  Hence  will  E  —  y  if  the  ex- 
pressions (12)  are  equal.  After  n  —  \  such  divisions,  the  quotients 
are  Qn-\  and  pj^x  —  A)  +  ^^^  and  E  will  equal  y  if  these  quotients 
are  equal.  Dividing  them  by  a;  —  A,  we  obtain  in  both  cases  the 
quotient  j»j)  and  the  remainder  r„.     Hence  E-^y, 

A  more  direct  proof  follows  from  the  remainder  theorem  (§  27). 
Assume  that  y  ^  E,  where  the  coefficients  r„,  .  .  .  ,  r^,  r^  of  E  are 
as  yet  undetermined.     Setting  x  =  h,  vfQ  get 

A^" +i^i  ^""' +  .  .  . +i^n-i/^  +  ;?n  =  ^. 
But  the  left  member  is  the  value  of  y  when  h  is  written  for  x  and 
is  therefore  the  remainder  left  on  dividing  ^  by  ^  —  h  (§27).  By 
inspection,  the  right  member  r^  is  the  remainder  left  on  dividing  E 
hy  X  —  h.  Since  the  two  remainders  are  equal  and  the  two  divi- 
dends y  and  ^are  assumed  to  be  equal,  the  quotients  must  be  equal: 

Qi  -  Poi^  -hr-'  +  rn{x  -hy -'  +  ...  +  r,{x  ^  h)  +  r,. 


Sec.  166]  COLLEGE  ALGEBRA,  173 

Proceeding  as  before,  the  division  of  Q^h-^  x  —  h  leaves  a  remainder 
equal  to  r^ ;  etc.  Hence  the  undetermined  coefficients  Vn,  -  *  - ,  r^,  r^ 
in  U  turn  out  to  have  the  values  given  in  the  theorem. 

166.  Kesuming  the  equation  y  =  u^  -{-  7u^  -\-  llu  --  2  of  §  164, 
where  ti  is  a  decimal,  we  seek  the  value  of  u  which  makes  y  —  0. 
The  powers  u^  and  v?  are  small  relative  to  llu  —  2.  Hence 
u  is  approximately  .1.  In  verification,  we  observe  that  y  is 
negative  for  u  =  ,1  and  positive  for  u  =  .2  and  hence  is  zero  for 
some  intermediate  value  (by  the  graph  or  by  the  formal  proof  in 
§  170).  Setting  7t  =  ,1  -\-  z,  the  equation  for  z  is  found  by  syn- 
thetic  division  (§  165)  as  follows: 

1      7  11  -  2         I J^ 

.1  .71  1.171 


1      7.1  11.71  ,    —    .829  =  first  remainder  r^ 

.1  .72 

1      7.2  ,  12.43  =:  second  remainder  r, 

.1 


1  ,    7.3  =  third  remainder  r^ 

.'.  y  =  z^  +  "T.dz^+UASz  -  .829,  a  result  identical  with  (10). 
Hence,  as  in  §  162,  z=  ,0Q  -\-  w,  where  w  is  of  the  denomination 
thousandths.  By  synthetic  division,  we  form  the  equation  (11) 
for  10 : 


1      7.3 

12.43           -  .829         .06 

V-.  . 

.06 

.4416            .772296 

1      7.36 

12.8716  ,    -  .056704 

.06 

.4452 

\ 

1      7.42  , 
.06 

13.3168                                       V\ 

1  ,    7.48 

y=w^  + 

7.48?(;2  _|_  13.3168^;  -  .056704. 

Then  ^  =  0  for  «^  =  .0042  -{-,  so  that  the  positive  root  is  1.1642  +. 


174        GRAPHIC  ALGEBRA;    NUMERICAL   EQUATIONS.       [Ch.  XVIII 

167.  For  a  negative  root  —  x  oi  x^  -\-  Ax^  —  7  =  0,  we  may 
solve  the  corresponding  equation  x^  —  4cX^  -[-7  —  0  with  the 
root  -f-  ^)  or  we  may  proceed  with  the  given  equation  by  the  usual 
method.  Thus  for  the  root  lying  between  —  3  and  ~  4,  we  set 
X  ==  —  4  +  ^,  where  t  is  positive  and  of  the  denomination  tenths 
and  proceed  as  usual.     The  work  is  exhibited  compactly  as  follows : 


4 

0 

~7|-4 

--  4 

0 

0 

0 

0 

-  7     |_j6 

-  4 

16 

-4 
~  4 

16 

•^    9 

-  8 

.6 

~  4.44 

6.936 

-  7.4 

11.56 

—    .064    .008 

.6 

-  4.08 

-  6.8      , 
.6 

7.48 

•*■  y 

-  6.2 

.008 

~     .049536 

.0594437 

-  6.1.92 

7.430464, 

-     .0045563     .000617 

.008 

-     .049472 

-  6.184, 
,008 

7.380992 

^    9 

-  6.176 

Hence,  to  six  decimal  places,  the  root  is— 4+. 608617=— 3.391383. 

It  is  seen  that  the  final  terms  16^  —  7  of  the  first  transformed 
equation  do  not  suffice  to  determine  t  very  accurately;  whereas  in 
the  later  transformed  equations  the  last  two  terms  determine  the 
figure  of  the  root  sought. 

Similarly,  the  third  root  is  —  1.7728,  to  four  decimal  places. 

\ 


I 


Sec.  168] 


COLLEGE  ALGEBRA. 


175 


168.  Consider  as  anew  example  the  equation  y=x^-\-&x'^-\-8x-\-S, 
whose  graph  is  given  in  Fig.  13,  liaving  been  traced  through  the 
points  (1,  23),  (.5,  13|),  (0,  8),  (-  .5,  5f), 
(-  1,  5),  (-  1.5,  6i),  (-  2,  8),  (-  3,  11), 
(-4,  8),  (-4.5,  2f),  (-5,  -7).  By 
accurate  plotting  in  the  vicinity  of  the 
two  bends  in  the  curve,  we  find  that  they 
occur  at  x  =  —  .845  and  x  =  —  3.155, 
approximately.*  The  corresponding  y- 
coordinates  are  4.9209  and  11.079,  respec- 
tively. The  fact  that  the  lowest  point  of 
the  bend  which  points  downwards 'is  above 
the  X'Uxi^  follows  more  simply  from  the 
form  y  z=z  (x  -\-  "^Y  —  4:X  in  which  the 
equation  may  be  written.  Hence,  for 
—  2  <  X  <  0,  X  -{-  2  and  —  ix  are  posi- 
tive and  therefore  also  y.  On  account  of  Fig.  13. 
the   double   bend,  the   graph  crosses  the 

:?;-axis  but  once;  for,  if  the  branch  below  the  :r-axis  should  bend  and 
cross  the  axis  again  (necessarily  at  the  left  ot  x  =  —  5),  or  if  the 
upper  branch  should  bend  and  cross  the  ic-axis  (necessarily  at  the 
right  of  X  =  +  1),  it  would  be  possible  to  draw  a  straight  line 
which  would  meet  the  graph  in  at  least  four  points.  To  show  that 
this  is  impossible,  we  note  that  the  equation  of  the  straight  line  has 
the  form  y  =  7nx  -{-  n.  The  ^-coordinates  of  its  intersections 
with  the  curve  are  given  by 

mx  -{-  71  =  x^  +  6x^  +  8:r  +  8. 

Hy  §  77,  this  equation  of  the  third  degree  has  at  most  three  roots. 

To  give  an  algebraic  proof,  we  observe  that,  for  :?;  <  —  6, 
:^  -f-  2  <  —  4,  so  that  the  condition  (x  +  ^Y  <  4^  will  be  satis- 
fied if   -  {x  +  2Y  <  X,   viz.,  if  -  (2;  +  f)2  <  -  (3)2^  which  is 


Y 

"o  T 


^  By  the  Cfalculus,  the  values  of  x  are  the  roots  —  2  ±  J  4/I2  of  the  equa- 
tion 3^2  4-  12aj  +  8  =r  0. 


1 


176       GRAPHIC  ALGEBRA;   NUMERICAL  EQUATIONS.      [Ch.  XVIIl 

true,  li  X  >  2,  then  i?;  -|-  2  >  4,  and  the  condition  {x  -f  2)^  >  4a? 
will  be  satisfied  if  {x  +  2)^  >  x,  viz.,  if  [x  +  If  >  J,  which  is 
true.  I 

169.  Descartes'  Rule  of  Signs.*  An  equation  f{x)  =  0  zvith  real 
coefficients  cannot  have  more  positive  roots  than  there  are  changes  of 
sigji  inf{x),  and  cannot  have  more  negative  roots  than  there  are 
changes  of  sign  in  f{—  x). 

For  example,  the  terms  of  f{x)  ~ -\-  x^  -{-  6x^  +  8^  +  8  =  0  all 
have  +  signs,  so  that  there  are  no  positive  roots;  there  are  three 
changes  of  sign  in  /( —  x)  =  —  x^  -^  6x^  —  8x  ~{-  8  =z  0,  so  that 
there  are  not  more  than  three  negative  roots. 

The  equation  f{x)  ^x^  —  6x^  —  r^-s  _|-  7:^;  _  2  =  0  has  at  most 
three  positive  roots,  and  at  most  two  negative  roots  since 

f{-x)  =  -x^-5x^  +  x^-'7x-2=zO 

has  two  changes  of  sign.  Hence  the  equation  must  have  at  least 
9  —  5  ^4  imaginary  roots  [see  §  181]. 

Suppose,  in  general,  that  the  positive  roots  of  f{x)  =  0  are 
^1,  fljg,  .  .  .  ,  «p.     Then /(a;)  is  divisible  (§  27)  by  the  product 

{x  —  a^{x  —  a^  ,  .  ,  {x  —  ap). 

If  the  quotient  is  Q(x),  the  roots  of  Q{x)  =  0  give  all  the  negative 
and  imaginary  roots  of  f{x)  =  0.  Let  the  signs  of  the  terms  of 
Q{x)  be,  for  example, 

+  +^_-  + +  -+. 

To  find  the  signs  of  the  product  {x  —  a^)Q{x),  we  employ  as  multi- 
plier a  binomial  with  the  signs  -| .     The  signs  of  the  products 

are  therefore 

Sum:  +±-q:+_qi:p  +  -4-- 

*  Descartes'  La  Geometrie,  Ley  den,  1637.  For  an  account  of  his  life  and 
work,  see  Ball,  History  of  Mathematics,  pp.  270-279. 


f 


Sec.  170]  COLLEGE  ALGEBRA,  177 

We  observe  that  in  the  product  an  ambiguity  of  sign  replaces  each 
continuation  of  sign  in  Q{x),  the  signs  before  and  after  an  ambiguity 
or  set  of  ambiguities  are  different,  and  a  change  of  sign  is  intro- 
duced at  the  end.  Hence  for  every  isolated  ambiguity  and  the  sign 
just  preceding  and  just  following  it,  the  number  of  changes  of  sign 
(whether  we  take  the  upper  or  lower  sign  in  the  ambiguity)  is  the 
same  as  for  the  three  corresponding  signs  in  Q{x).  An  analogous 
result  holds  for  a  set  of  ambiguities  if  we  take  all  the  ambiguities  in 
the  set  to  be  alike.  But  if  the  latter  are  not  taken  alike,  there  is  one 
or  more  additional  changes  of  sign  in  the  product  than  in  Q{x), 
There  is  always  an  additional  change  of  sign  at  the  end.  Hence 
the  product  contains  at  least  one  more  change  of  sign  than  Q{x), 
After  multiplying  in  the  p  factors  x  —  a^,  x  —  a^  ,  ,  .  .  ,  x  —  a^, 
we  obtain /(x),  which  therefore  has  at  least  ^  more  changes  of  sign 
than  Q{x),    Hence /(a;)  has  at  least  jy  changes  of  sign.     The  number 

of  positive  roots  is  thus  '^  c,  c  being  the  actual  number  of  changes 

of  sign  in  f(x). 

The  negative  roots  of  f(x)  =  0  are  the  positive  roots  of 
f(^—x)  =  0,  so  that  the  number  of  the  former  is  less  than  or  equal 
to  the  number  of  changes  of  sign  in/(—  x). 

Thus  x^  —  f^x^  -^  x-\-l  =  0  has  at  most  two  positive  roots  and 
at  most  two  negative  roots  and  consequently  has  at  least  two 
imaginary  roots. 

.  170.  Consider  the  graph  of  y  =f(x),  where  f{x)  is  a  rational 
integral  function  of  x  (§  76).  H  there  be  two  points  of  the  graph, 
one  of  which  is  above  the  x-^xis  and  the  other  below,  it  may 
be  proved  that  the  graph  must  cross  the  x-^x\s  at  one  or  more 
intermediate  points.  Thus  if  f{a)  is  positive  and/(Z>)  is  negative, 
there  is  some  value  of  x  intermediate  to  a  and  h  for  which  f(x)  is 
zero.  The  proof  consists  in  showing  that,  as  x  changes  gradually 
from  a  to  ^,  f{x)  also  changes  gradually  and  assumes  all  interme- 
diate values  between /(a)  and/(5),  so  that  in  passing  from  a  posi- 


1 


178       GRAPHIC  ALGEBRA;    NUMERICAL   EQUATIONS,      [Ch.  XVIII 

tive  to  a  negative  value  it  passes  through  the  value  zero.  While 
this  is  readily  proved  (see  example  below)  when /(a;)  is  of  the  form 

x""  +pX-'^+  .  .  .  +i?n, 
the  case  under  consideration,  it  is  not  true  for  all  functions  /(^). 
For  example,  it  fails  for  the  functions  y  —  2\/x  and  y  —  \/li)x  —  x^ 
whose  graphs  were  constructed  in  §  156,  since  there  are  two  values 
of  y  corresponding  to  each  value  of  x. 

Consider  the  example  f(x)  =  x^  -\-  4:X^  —  7.  As  x  changes  from 
a  to  a  +  /^^  f{^)  changes  from  f(a)  to /{a  +  ^0'  ^^^  increase  being 

{ (^  +  /O'  +  4(a  +  7^)2  _  7  [  -  ]  a^  +  4«2  _  7  [  | 

=  (3a2  -f.  Sa)h  +  {3a  +  4)¥  +  h\  ' 

If  h  is  small,  this  difference  is  small,  and  by  taking  h  sufficiently 
small,  this  difference  can  be  made  as  small  as  we  please.  Hence  if  x 
obliges  by  small  amounts  from  a  to  b,  f{x)  will  change  by  small 
amounts  from/(^)  to  f{b). 

Moreover,  a  curve  joining  two  points  which  lie  on  opposite  sides 
of  the  :r-axis  must  cross  that  line  an  odd  number  of  times;  but 
m!ret  cross  it  an  even  number  of  times,  or  not  at  all,  when  the  points 
lie  on  the  same  side  of  the  ^-axis.     We  may  state  the  theorem: 

If  f{x)  he  a  rational  integral  function  with  real  coefficients,  and 
the  values  f{a),  f{h)  given  ly  siihstituting  real  numbers  a,  b  for  x 
are  of  opposite  signs,  the  equation  f{x)  —  0  ?ias  an  odd  number  of 
real  roots  lying  betweeyi  a  and  b;  but  has  an  even  number  of  real 
roots  or  no  real  root  lying  betiueen  a  and  b,  if  f{a)  and  f{b)  are  of 
like  sign. 

Example  1.  Every  equation  of  odd  degree  has  at  least  one  real  root  of 
sign  opposite  to  that  of  the  constant  term. 

Thus  x^  -f  ^^^  —  7  =  0  was  seen  to  have  a  root  between  +  1  and  +  2. 

Let  the  equation  be /(a;)  =  x^  -\-  aa?'^-  ^  -\-  ,  .  .  -\-  c  =  0,  and  let  x  approach 

the  values  —  00 ,  0,  and  -f  go  .     The  sign  of  /(O)  is  the  same  as  that  of  c.     To 

find  the  sign  of/(±  oo),  divide /(ic)  by  a!"-i,  which  is  positive  since  7i  —  1  is 

h  c 

even.     The  quotient  x  -\-  a  -\-       +  •  •  •  ^ — ^rri  differs  from  ic  by  a  function 

which  approaches  zero  as  x  approaches  ±  00 .     Hence  f{-\-  00  )  is  positive  and 


I 


Sec.  170]  COLLEGE  ALGEBRA,  179 

f^— oo)is  negative.     Hence,  if  c  is  positive, /(ir)  changes  its  sign  between 
a?  =  —  00  and  x  —  0  and  therefore /(a;)  =  0  has  a  negative  root.     If  c  is  nega- 
:  tive,  f{x)  changes  its  sign  between  x  =  0  and  x  =  -\-  a:^ ,  ^o  that/^a?)  =  0  has  a. 
:  positive  root. 

Example  *2.  Every  equation  of  even  degree,  whose  constant  term  is  nega- 
tive, has  at  least  one  positive  and  at  least  one  negative  root. 

Upon  substituting  —  00,  0,  +  00  for  x,  the  results  are -f  cjo ,  c,  -j-  00 ,  re- 
spectively, w^here  c  is  negative.  Hence  there  is  a  real  root  between  —  00  and 
0,  and  a  real  root  between  0  and  +  ^  • 

(EXERCISES. 
1.  Plot  the  graph  of  y  -  x^  -{-  2x  —  \  and  solve  iu^  +  2«  —  1  =  0. 
2.  Plot  y  =  a^  -\-  2x  -\-  20    and    find   to  five    decimals    the    root    of 
V-f--   2x  +  20  ==  0  such  that  -  8  <  a?  <  -  2. 

3.  Plot  y  =  x^  -{-  X  —  d  and  find  to  five  decimals  the  root  1  <  oj  <  2. 

k4.  Plot  y  =  x^  —  4lX,     What  are  the  roots  of  x^  —  Ax  =  0 '^ 
6.  Show  that  the  graph  of  y  —  x^  -]-  Sx^  -\-  2x  -^  1  crosses  the  aj-axis  but 
once. 

16.  Find  the  roots  between  1  and  2,-1  and  —  2,  of  x^ -^  dx"^  -  2x  -  6  =  0. 
7.  Find  the  root.between  2  and  3  of  a!^  +  dx^  -  80^  -  18  =  0. 
8.  Find  the  real  roots  of  a^  -{-  2x^  —  x  -  1  =  0. 
9.  Find  the  real  roots  of  «*  +  a;^  —  a;  —  2  =  0. 

10.  Find  the  real  roots  of  x^  +  4x^  -j-  Qx^  -  Sx  -  i  =  0, 

11.  If  the  terms  of  an  equation  are  all  positive,  it  has  no  positive  root. 

12.  If  the  terms  of  an  equation,  with  all  powers  of  x  present,  are  alter- 
nately positive  and  negative,  it  has  no  negative  root. 

13.  The  equation  x^  -f  a^x  -{-  b'^  =  0  has  one  negative  root  and  two  imagi- 
nary roots. 

14.  Ifn  is  even,  a?^  -J-  1  =  ^  has  no  real  root ;  if  7i  is  odd,  —  1  is  the  only 
real  root. 

15.  The  equation  a^  -\-  2x^  -\-  dx  —  1  =  0  has  at  least  six  imaginary  roots. 
16i  x^  -\~  12ic^  -\-  6x  —  9  =  0  has  a  positive  root,  a  negative  root,  and  two 

imaginary  roots. 

17.  For  n  even,  ir^  —  1  =  0  has  only  two  real  roots  ;  for  n  odd.  only  one. 

18.  aj^  —  7a;*  —  3«2  +  7  =  0  has  at  least  two  imaginary  roots. 


CHAPTER  XIX. 
THEORY  OF   EQUATIONS. 

171.  The  solution  of  the  general  quadratic  equation  (see  §  31) 
was  known  to  the  Arabians  as  early  as  the  ninth  century.  The 
solution  of  the  cubic  equation  x^  -\-  qx  -\-  7'  =  0  was  discovered  by 
Scipio  Ferreo,  who  imparted  it  to  his  pupil  Florido,  in  1505.  The 
latter  proclaimed  his  knowledge  of  the  solution  upon  learning  that 
Tartaglia  had  solved  the  analogous  equation  x^  -f  mx'^  -f  ^  =  0,  But 
Tartaglia  doubted  the  truth  of  his  statement  and  challenged  him 
to  a  trial  in  the  year  1535,  having  succeeded  in  rediscovering  Ferreo's. 
solution.  Then  Cardan  appeared  and  solicited  from  Tartaglia  his 
rules  for  solving  various  cubic  equations  and  finally  succeeded  in 
getting  them  after  giving  the  most  sacred  promises  of  secrecy. 
But  in  1545  Cardan  broke  his  promises  and  published  the  rules  in 
his  Ars  Magna.  .  Tartaglia  had  intended  to  publish  his  rules  in  his 
own  work,  but  died  in  1559  before  the  part  on  cubic  equations  was 
reached,  so  that  no  mention  is  made  by  him  of  his  discoveries.  In 
time  the  rules  came  to  be  regarded  as  the  discovery  of  Cardan  and 
to  bear  his  name. 

172.  Consider  the  cubic  equation  in  the  most  general  form 

(G)  x^  +  ax^  +  hx  +  c=.0. 

Setting  a;  =  y  — —,  the  equation  becomes 

o 

(R)  f+[^-YF  +  ['-T  +  ^)  =  ''' 

II  i8o 


Sec.  172]  COLLEGE  ALGEBRA.  l8l 

an  equation  lacking  the  square  of  the  unknown  quantity,  and  called 
the  reduced  form  of  equation  (G).  It  suffices  to  solve  the  equation  in 
y,  since  the  values  of  x  are  then  given  by  the  relation  x  —  y  —  a/3. 

Consider  then  the  reduced  cubic  equation 
(1)  ^  ^'+i^i/  +  ^==0. 

Making  the  substitution 


(2) 

y  =  z- 

P 

3«' 

the  cubic 

(1)  becomes 

,8         P' 

+  r  = 

=  0. 

Multiplying  by  2;^ 

the  latter  becomes 

(3) 

2-6    ^  ,.j;3  _ 

27 

0. 

Considering  z^  to  be  the  unknown  quantity,  equation  (3)  is  a  quad- 
ratic equation,  the  solution  of  which  gives  z^  and  then 

(4)  z  -  y^-  '-  ±  ^R,  R  =  '^+^- 


Kemembering  that  a  quantity  q  has  three  cube  roots  *  (§8), 

l/'q,^       00  p'q,^       od'^  l/q'  (ct)  =  —  |  +  |  /  -  3,  cy^  —  1) 

it  would  seem  that  the  six  values  of  z,  given  by  (4),  would  lead  to 
six  roots  y,  given  by  (2),  of  the  cubic  equation  (1).     But 

(-i+^s)(-;--*'^)=T-^=-4 

Hence,  if  the  notation  of  two  of  the  cube  roots  (4)  be  definitely 
fixed  in  such  a  manner  that 


*  The  vertical  bar  at  the  right  of  the  radical  sign  is  used  to  mark  the 
choice  of  a  particular  root. 


I«2 


THEORY  OF  EQUATIONS, 


[Cn.  XIX 


then  the  remaining  four  cube  roots  (4)  may  be  paired  so  that  the 
product  of  the  two  in  each  pair  is  —  i?/3,  viz., 


V-i 


p 


+  i/R     .  oo^!/-L-^R     =_-|, 


'lJ-\  +  VR    .-^- 


^=-f 


But  the  vahie  of  y  given  by  (2)  remains  unaltered  if  we  replace  z 
by^;  mdeed. 


3z        'd    '    [  dz  J 


'dz' 


Hence  the  two  values  of  z  in  any  one  of  the  three  pairs  give  the 
same  value  to  y,  and  the  value  of  y  equals  the  sum  of  the  two^ 
values  of  z  in  that  pair.     Hence  the  three  roots  y  are 


(5) 


3  /_  r         /—  '       ^  I  —  r         /- ' 


Y-^^^'-'-'v     2 


/i^, 


f  |/ii  +  CB*  /  -^  -   ^E. 


2/3- -A/      3 
Example.     Solve  the  reduced  cubic  equation  ^^  _  15^  _  126  =  0. 

Set  V  =  2  —  (^^^]  =  2  +  -.     Then  z^  —  12Qz'  +  135  =  0. 
\   Sz  J  2 

...      (23  _  1)(23  -  125)  =  0. 

• .     2  =  1    fij,  oi)^,  5,  5(»,  or  5(»^     Then  y  =  z  -^  —  equals  respectively 

'  z 


6     00  H =  GO  -r  5gi3^     g?^  +  5GJ, 

GD 


5gi?  -|-  gi?^,     5gi3'  -|-  go, 


Sec.  173]  COLLEGE  ALGEBRA,  183 

giving  only  three  distinct  roots  y.     Since  ca^  -[-  &?  -f  1  =  0  (§  8),  they  are  6, 
—  4a3  —  5,  4o!9  —  1.     Since  go  =  — i  +  ii^  —  3,  they  become 


6,     _3_2|/-3,       -3-|-2|/-3. 
The  two  imaginary  roots  are  conjugate  (§  7). 

173.  Formulae  (5)  for  the  roots  are  known  as  the  formulae  of 
Cardan.  The  method  used  for  obtaining  them  is  essentially  that 
given  by  Hudde  in  1650. 

According  to  the  value  of  R^  —  +  ~,  we  have,  when p  and  r 

are  real,  for  R  posiiive,  one  real  root  and  tivo  conjugate  imaginary 
roots  ;  for  R  equal  zero,  all  roots  real  and  two  of  them  equal;  for 
R  negative^  all  three  roots  real  and  distinct. 

In  proof,  suppose  first  that  7?  >  0.  Let  the  radicals  be  A  and 
B  so  that  the  roots  are  A-\-  B,  ooA  -\-  od^B^  od^A-{-  goB.     Since 


'=-i+i4/-3,         G92^  -i-i4/-3, 


the  three  roots  are   A  +  B  and  -  i{A  +  B)  ±i{A-  B)\/  -  3. 
Since  A  and  B  are  now  real,  one  root  is  real  and  the  others  are  con- 
jugate imaginaries. 
I       li  R=0,  the  roots  are  A -{- B,- ^{A -^  B),  -  \{A  +  B),  and 
hence  are  all  real. 

If  i?  <  0,  the  radicals  in  formula  (5)  are  imaginary.     But  (see 

Appendix)    there   exists   a  cube  root  of  — -  +  \/R  of  the  form 


a  +  ^\/  —  I,  a  and  /5  being  real,  the  other  two  cube  roots  being 


QD{a  -\- 15\/  —  1)  and  Go^[a  +  /?|/  —  1).  The  conjugate  imaginary 
—  -  —  \/ R   will    then    have    as    its    cube    roots    a  —  (3  \/  —  1,  , 

fd^[a  —  /Jj/—  1),  GD{a  —  /3\/  —  1).  Hence  the  roots  (5)  may  be 
written  (since  the  product  of  the  two  terms  in  any  root  is  the  real 
number  —  ^/3): 


1 84  THEORY   OF  EQUATIONS,  [Ch.  XIX 

(a  +  /^|/^=3)  +{a-  Ps/~^\)  =  2^, 
G?(ar  +  /?!/-  1)  +  G^(ar  -  p\/~^^\)  =  -  a-  j3\/3, 

These  real  roots  are  readily  seen  to  be  unequal. 

Thus  when  the  roots  of  a  cubic  equation  are  all  real  and 
unequal,  Cardan's  formulae  present  their  values  in  a  form  involving 
imaginaries.  In  eliminating  these  imaginaries,  it  was  necessary  to 
express  the  cube  root  of  a  complex  number  in  the  form  of 
a,-{-  ^  ^  —  \,  If  we  attempt  to  do  this  algebraically,  we  find  that 
a,  /?  depend  upon  equations  similar  to  the  proposed  equation.  For 
example,  a  depends  upon  a  cubic  equation  all  of  whose  roots  are 
real,  so  that  its  roots  would  depend  upon  expressions  as  complicated 
as  a  itself.  Hence  the  case  when  all  three  roots  are  real  is  called 
the  '* irreducible  case,"  since  Cardan's  formulae  then  have  no  prac- 
tical value.  But  this  case  is  precisely  the  one  which  can  be  readily 
solved  by  trigonometry.    [See  Appendix.] 

174.  Since  gj^  -{-  oo  -\-  1  =  0,  we  find  by  formulae  (5)  that 

2/i  +  !/2  +  !/3  -  (1  +  ^  +  ^%^  +  £)  =  0, 
y^y^Vz  =  {^  +  ^)(^'  +  ^'  +  oj'^B  +  gdAB)  =  A'  +  B^=^r, 


3  /  _  r         ,—              ^  I  —  r         — ' 
where    we    have    set    J[  =  a  /  -— 1-  y  ^,     ^  ~  \  /  -^ \/R. 

Hence  in  a  cubic  equation  lacking  the  term  y^,  the  sum  of  the 
roots  is  zero,  the  sum  of  the  products  of  the  roots  taken  two  at  a 
time  is  the  coefficient  of  y,  and  the  product  of  all  the  roots  is  the 
negative  of  the  constant  term. 


I 


Sec.  175]  COLLEGE  ALGEBRA,  185 

EXERCISES. 

Solve  the  cubic  equations 

1.  a?  -  l%x  +  35  =  0.  2.  x^  -  15a;  +  4  =  0. 

3.  0!^  -  27aj  +  54  =  0.  4.  x''  -f  6:r«  +  3a;  +  18  =  0. 

5.  28ar*  +  9a;2  -  1  z=  0.  6.  a;^  _|_  4^2  _  ^^j  +  6  =  0. 

7.  Since  a;,  =  .Vi  —  a/3,  x^  —  y^  ^  a/3,  aJs  =  ^3  —  a/3  are  the  roots  of 
the  general  equation  (G)  if  y, ,  ^2 ,  ys  are  the  roots  of  the  reduced  equation 
(R),  show  that  aJi  +  a^a  +  a^s  =  —  a,  a^iaja  +  a^ia^a  +  X'^x^  =  b,  XiX^Xz  =  —  c. 

175.  The  solution  of  equations  of  the  fourth  degree  was  at- 
tempted by  Cardan,  but  was  first  accomplished  by  his  pupil  Ferrari. 
It  is  sometimes  ascribed  to  Bombelli,  who  published  it  in  his 
algebra  in  1579.  Other  solutions  have  been  given  by  Descartes  in 
1637  and  by  Euler  in  1770  and  by  many  later  writers. 

176.  FerrarVs  solution  of  the  quartic  equation 

(6)  a;^  +  px^  -{-  qx^  -{-  rx-\-  s  =  0. 

The  equation  may  be  written 

{7?  +  |:rj   =  {^^  -  q^x^  ^rx--s. 

In  case  the  second  member  is  not  a  perfect  square,  add  to  each 
member  [7?  +  ^Ay  +  -f--     Then 

(7)(.^+f.+i;=(.+^-.>'+(f-r).+(i^-.). 

The  second  member  will  be  a  perfect  square  (§  31)  if  ■ 

Hence  y  must  be  a  root  of  the  cubic  equation 

(8)  y^  —  qy'^  +  {'pr  —  45)y  +  ^^^  —  jf'S  —  r^  =  0, 

Let  y^  be  a  root  of  (8)  and  set  2/j,+  ^  •—  ^^  =  1".     The  right  mem- 


iS6  THEORY  OF  EQUATIONS,  [Ch.  XIX 

ber  of  (7)  may  be  written 

^L'+Kf---)/^]* 

Hence  (7)  factors  and  gives 

(9)  \ 

Solving  these  quadratics,  we  obtain  the  roots  x^,  x^;  x^,  x^  of  (6) 
We  observe  (§  32)  that 

Hence 

^1  +  ^2+^3+^4  =  -P^     ^,^2^,^,  =  -^  I  ^i'  -  \^~  -  rj  ~  T^  =8, 

^1^2 +  ^1^8 +  -^1^4 +  ^2^8 +  ^2^4  + ^3^4  ^^  ^1^2+ (^1  +  ^2/ (^3  +  ^4) +^3^4  ^^  S'' 
XJX20!yo'~\*^i^2l~\        1    Z    i~i~     2    Z    A  12x3  "I  4/    ~t~  ^^^a\^i  ~\      ^2)    ~~~ 

For  a  general  quartic  equation  (4),  the  sum  of  the  roots  is  —  p^ 
the  sum  of  the  products  of  the  roots  two  at  a  time  is  q^  the  sum  of  the 
products  three  at  a  time  is  —  r,  the  product  of  the  roots  is  s. 

Example.     Solve  the  quartic  equation  x^  +  2a;'  —  12ir^  —  10.^  -f  3  =  0. 
The  auxiliary  cubic  (8)  is  here  y^  -f-  12^^  —  32y  —  256  =  0,  one  of  whose 
roots  is  —  4.     Taking  y^  =  —  4,  we  get  T  =  9.     Hence  equations  (9)  are 

x^  -\-  Ax  -  1  -  0,        x^  -  2x  -  d  =  0, 

whose  roots  are  —  2  ±  |/5    and    3,-1. 

P 
177.  If  we  set  rr  =  2;  —  -^  in  (6),  we  obtain  for  z  a  quartic  equa- 
tion lacking  the  term  z^.     Consider  then  the  reduced  form 

(10)  z^+Qz^+Ez  +  S=^0. 


Sec.  178]  COLLEGE  ALGEBRA.  187 

Descartes'  solution  consists  in  factoring  tlie  left  member  of  (10) 
into 

(z^ -\-lcz^  l){z^  -Jcz+  m)  =  z^+{l  +  m-  k^)z^  +  h(m  -  l)z  +  Im. 

In  order  that  this  expression  shall  be  identical  with  the  left  member 
of  (10),  the  corresponding  coefficients  must  be  equal: 

Z  +  m  -  F  =  Q,     ]c{m  -I)  =  E,     Im  =  S. 

Substituting  the  values  of  I  and  in  derived  from  the  first  two  rela- 
tions into  the  third  relation,  we  get 

{Q  +  Jc^-\-B/k){Q-\-]c'- R/1c)=^4.S, 

We  have  thus  a  cubic  equation  for  P.  When  it  is  solved,  we  derive 
the  corresponding  values  of  I  and  m.  Then  the  roots  of  (10)  are 
given  by  the  solution  of  the  two  quadratic  equations 

z^+kz  +  l^O,     z^-kz  +  m  =  0. 

For  example,  to  solve  z*  —  'dz^  +  6;^  ~  2  —  0,  we  have  the  con- 
ditions 

/  +  m  -  F  ^  -  3,     k{?n  -  /)  =  6,     Im  =  -  2. 

.  •.     k^  -  6k*  +  17^-  -36  =  0. 

The  latter  is  satisfied  by  k^  =  4.  Taking  k  =  2^  we  get  m  —  2, 
Z=  -1.       • 

...     z*  ^^z^  +  6z-2  =  {z^  +  2z-  1)(^2  _  2^  +  2). 

The  roots  are  therefore  —  1  ±  |/2  and  1  ±  |/—  1. 

178.  Elder's  solution  of  tlie  quartic  equation  (10)  consists  in 
assuming  as  the  general  expression  for  a  root 

Squaring  and  transposing  the  terms  free  of  radicals,  we  get 
z^  —  u  —V  —  tv  —  2(|/w  4/^  +  V'^^  V'^  +  V'^  V^^  )• 


1 

1 88  ,  THEORY  OF  EQUATIONS,  [Ch.  XIX 

Squaring  the  latter  and  reducing,  we  get 

z^  —  2{u-\-v-\- 7v)z^  —  s{u]/v  \/w  +  v\/u  \/w  +  2Vi/u  i/v)-{-A  =  0, 
where  A  =  {u  -{-  v  -\-  wY  —  4:{tiv  +  tiw  +  vw). 

,\     z'^  —  2{u  +  V  +  iv)z^  —  Sz  \/u  \/v  \/tv  +  ^  =  0. 
Comparing  the  latter  with  equation  (10),  we  get 

u+  V  -{-  w  =  —  IQy     \/u  \/v  \/w  —  —  \R,     A  —  S. 

Hence,  by  Exercise  7  of  §  174  or  by  the  theorem  of  §  182,  w,  v,  w 
are  the  roots  of  the  cubic  equation 

t'  +  W  +  {M'  -\s)i-  i^i^  =  0. 

Since  the  expression  for  z  involves  three  square  roots,  it  might 
seem  that  the  quartic  equation  would  have  eight  roots.  But  the  signs 
of  the  square  roots  must  be  chosen  so  that  4/?^  \/v  \/w  =  —  IE,  so 
that  only  two  signs  are  arbitrary,  giving  four  roots. 

EXERCISES. 

Solve  the  quartic  equations 
1.  2*  -  22^  _  82  -  3  =  0.  2.  x^  -  2x^  -  W  +  8aj  +  12  =  0. 

3.  2*  -  IO22  -  2O2  -  16  =  0.  4.  a^  -  Sx^  +  9x'  -{-  Sx  -  10  =  0. 

5.  If  a  quartic  has  two  equal  roots,  the  auxiliary  cubic  has  two  equal  roots, 
and  inversely. 

6.  If  a  quartic  has  two  distinct  pairs  of  equal  roots,  then  two  of  the  roots 
of  Euler's  auxiliary  cubic  are  zero. 

7.  The  roots  of  the  quartic  (6)  are  l{- p  ±  |/Z  ±  i^B  ±  |/c),  all  the 
signs  being  -f  ,  or  two  signs  —  and  one  sign  -f-  ,  if  ^,  i?,  C  are  the  roots  of 
W^  +  {Sq-dp^)  W  +  {Zp'^-X^p'q  +  16^2  _|_  jg^^  _  ^4^)  ^_  (^3_4^^  _|_  g^^2  ^  q. 

8.  Solve  the  general  cubic  by  multiplying  in  a  new  factor  x~\,  and  using 
A  to  make  the  auxiliary  cubic  solvable  by  inspection. 

179.  During  the  eighteenth  century  many  unsuccessful  attempts 
were  made  to  solve  the  general  equations  of  the  fifth  and  higher 
degrees.  In  1770  Lagrange  analyzed  the  methods  of  his  predeces- 
sors and  traced  all  their  results  to  one  principle  and  proved  that  the 


Sec.  179]  COLLEGE  ALGEBRA.  189 

general  qiiintic  equation  cannot  be  solved  in  this  way,  since  the 
auxiliary  equation  is  of  the  sixth  degree  and  of  essentially  general 
character.  It  was  then  proved  *  by  Abel,  Wantzel,  and  Galois  that 
it  is  impossible  to  solve  by  algebraic  means  the  general  equation  of 
degree  n  for  n  >  4:.  But  the  equation  of  the  fifth  degree  may  be 
solved  in  terms  of  elliptic  functions  f  (Hermite).  The  roots  of  cer- 
tain equations  of  the  fifth  degree  may  be  expressed  by  radicals.  X 

Example  1.  Find  all  the  roots  of  ^  —  1  =  0. 
The  four  j-oots  different  from  the  root  1  satisfy 

^"""  \   =  a;*  +  a^  4-  a;2  +  a?  +  1  =  0. 
X  —  1 

If  we  divide  by  01^  and  set  a?  -| =  y*  we  find  that 

...     a.'^_|-(-l  ±  |/5")  +  l  =  0. 
Hence  there  are  four  imaginary  fifth  roots  of  unity  : 


i{_l-p  |/5  ±  |/l0-|-2|/5   i/-l!,      il-1-  1/5  ±|/lO  -  2|/5  V  -^\ 

If  we  denote  one  of  them  by  e,  the  others  are  e^,  e^,  e*.  Indeed,  since  e^  =  1, 
then  {e'^f  =  1,  (e^)^  =  1,  {e^f  =  1.  Also  no  two  of  them  are  equal;  for  ex- 
ample, e  =  e^  gives  e  —  ±  1,  e  =  e*  gives  e^  —  e'^  =  1.     We  notice  that 

1  +  e  +  6-2  +  e^  4-  e*  =  0,     l-e-e^'e'^'e*  =  e^o  =  1. 
Example  2.  Solve  the  equation 

^  -]-pf  +  Wy  +  ^  ==  0. 

Substituting 

(11)  y=^'-  S' 

*  See  Serret,  Cours  d'Algebre  Superieure,  t.  2,  sec.  Y,  ch.  II. 

f  On  this  and  allied  subjects,  see  Jordan,  Traite  des  substitutions  et  deS 
equations  algebriques,  Paris,  1870  (pp.  370-382). 

if  Concerning  solvable  quintics  see  McClintock,  American  Jour,  Math., 
Vol.  6,  p.  301. 


190  THEORY  OF  EQUATIONS.  [Ch.  XIX 

the  equation  becomes 

We  therefore  obtain  an  equation  of  the  second  degree  for  z*.     Hence 
Proceeding  as  in  §  173,  we  obtain  the  five  roots 


Y  -  y  +  v«  +  e^-'\/-  f  -  i^e       a  =  0, 1, 2, 3, 


4), 


where  e  denotes  an  imaginary  fifth  root  of  unity  (Example  1). 

The  same  method  is  to  be  followed  in  working  Exercises  9-13  below,  the 

substitution  being  y  —  z  —  ~  for  an  equation  of  degree  n,  as  in  (2)  and  (11). 

Example  3.  The  equation  y^  -f  py^  -f  qif'  -\-  qy'^  -\- py  ^  \  —  0  may  be 
solved  by  algebra.  It  has  the  root  —  1.  Dividing  out  the  factor  y  -f  1>  we 
obtain  the  quartic  equation 

y'-\-{v-  1)^/'  +  (?-;?  +  i)y'  +(^-  i)y  +  1-0. 

The  latter  may  be  solved  by  the  earlier  methods  or  more  simply  by  setting 
y  A —  z,  whence  y'^  A z=z  z'^  —  2,  so  that 

y  y 

z^A^{p-\)z-^q-p-l  =  0. 


Then  y  =  \{z  -\-  \^z^  —  4)  gives  the  four  values  of  y. 

180.  A  reciprocal  equation  is  one  which  is  not  altered  by  re- 
placing the  variable  by  its  reciprocal.  The  equations  solved  in  ex- 
amples 1  and  3  of  §  179  are  reciprocal  equations.  Just  as  these 
special  equations  were  solved  by  means  of  equations  of  lower  degree, 
so  the  solution  of  any  reciprocal  equation  may  be  made  to  depend 
upon  the  solution  of  an  equation  of  lower  degree. 

If  the  given  equation  is 

(12)       y^  ^p^yn-^J^p^  yn-^  +  .  .  .  +  Vn-^V^  +Pn-l  V  +Jt?„  =  0, 


I 


Sec   180]  COLLEGE  ALGEBRA,  191 

the  equation  obtained  by  writing  —  for  y  and  multiplying  by  3/"  is 

If  the  two  equations  are  to  be  the  same,  we  must  have 

PnVx^Vn-X.pnl\-Vn-'l,   '   •    •    ,  PnPn  -  2="  P^  ^  PnPn -I  ^  P^  ,  Pn     =  ^^ 

Hence  jt?„  =  ±  1,  so  that  there  are  the  two  following  types: 

(I)  Pn^+^,       Pl=Pn-l         P^'-^Pn-%y       P^=Pn-Z,   '   '    ', 

(II)  J0„=-1,     P,^-Pn-l,     P^——Pn-%,     Pz=    -  Pn-Zy   y" 

P         Suppose  that  (12)  is  a  reciprocal  equation  of  type  (I).     Then 
(13)  (r+l)+^,(i/^-^+2/)+i^.(2/"-'+^')+it^3(!/"-*^+^')+  .  .  .  =0. 

If  n  is  odd,  7i  =  2t  -{-  1,  the  hnal  term  in  (13)  is  Pt{y^'^^  -f-  y^). 
The  terms  are  all  divisible  hjy-\-l  and,  if  the  quotient  be  Q{y)y 
the  equation  Q{y)  =  0,  of  even  degree  tj  —  1,  is  a  reciprocal  equa- 
tion. In  fact,  the  dividend  (13)  and  the  divisor  y  -\-  1  remain  un- 
altered, aside   from  a  factor  ?/"*   and  y,  respectively,  wlien  —   is 

written  for  y,  so  that  a  similar  result  must  hold  true  for  the 
quotient  Q{y).  Moreover,  Q{y)  =  0  is  of  type  (I)  since  the  con- 
stant term  is  +  1,  being  derived  from  the  division  of  ?/"  -f  1  {n  odd) 
by  ^  +  1  [see  formula  (3),  Chapter  III].  As  an  illustration  see 
Example  3  above.  Hence  Q{y)  =  0  is  of  the  character  next  treated. 
I  If  71  is  eveTi,  n  =  2t,  the  final  term  in  (13)  is  p^  y*.  Dividing 
the  equation  by  y^,  we  get 

{y'+^-)+p{y'-'+^)+ply'-'+^^+-  ■  ■+p.-^{y+l)+P.=o. 

8ety  +  ^=z.  Then  ^^  +  i  =  ^^  _  2,  yB_^l_^^s_  3^,  rj,^^ 
general  term  may  be  derived  from  the  identity 

(14)    r+y  =  <r-'  +  ^.)  -  (r-  +  5^.). 


J9^  THEORY  OF  EQUATIONS.  [Ch.  XIX 

•*•     y^  ~\ 3"  ==  ^(^^  —  2)  —  z  =  z^  —  3z. 

1_ 
.By  mathematical  induction  (an  exercise  left  to  the  reader),  we  get 


y^  +  -j  =  z{z^  -  4.z^  +  2)  -  (z^  -  Sz)  =  z^  -  5z^  +  6z. 


m(m-^-l)(m-;?-2)...(m-2;7+2)(m-2^+l)^^_, 
i-^     ij  1.2.3...^  ^       +••• 

Hence  equation  (13)  is  reduced  to  an  equation  of  degree  t  =  n/2. 

Suppose  next  that  (12)  is  a  reciprocal  equation  of  type  (II).     If 
n  is  odd,  n  =  2t  -{- 1,  the  equation  may  be  written 

(r-1)  +P,{r-'-y)  +P,{y''-'-y')  +  • . .  +Pt{y'^'-y')  =  o, 

and  the  left  member  is  divisible  by  y  —  1.  Calling  the  quotient 
Q{y)y  we  note  that  the  dividend  and  divisor  remain  unaltered,  aside 

from  the  respective  factors  —  «/"  and  —  y,  when  -  is  written  for 

y,  so  that  a  similar  result  must  hold  for  Q{y).  Hence  Q{y)  =  0  is 
a  reciprocal  equation  of  even  degree  n  —  1.  Its  constant  term 
is  4-  Ij  since  it  arises  from  the  division  of  y''  —  Ihy  y  —  1  [see 
formula  (2),  Chapter  III].  Hence  Q{y)  =  0  is  of  type  (I),  whose 
reduction  to  an  equation  of  half  the  degree  was  just  effected. 
If  n  is  even,  7i  =  2t,  the  equation  becomes 

{y-^l)-\-p,{y--'-y)+p,{y--'-y')  +  ,..  +p,_,{y^  +  ^-y^-^)  ^ o, 

and  the  left  member  is  divisible  by  y^  —  1.  Dividing  by  this  factor, 
we  obtain  an  equation  of  even  degree  n  —  2,  which  may  be  shown, 
as  in  the  previous  case,  to  be  a  reciprocal  equation  of  type  (I). 


Sec.  181]  COLLEGE  ALGEBRA,  193 

In  addition  to  the  examples  1  and  3  above,  consider  the  equation 
/  -  1  =  0. 
Eemoving  the  factor  y'^  —  1,  we  get  the  reciprocal  equation" 

Setting  y  -\ =  ;2,  we  get  ^^^  —  1  =  0.     Hence  ^  +  -  =  ±  1,  or 

2/'-2/  +  l  =  0,     ^2  +  ^^  +  1  =  0. 
/example.  Solve  ^  -  5y*  +  9^^  -  9^^  _|_  5^  _  1  ^  0. 
Dividing  out  y—1,  we  get  y^—^y^  +  5y^—4:y  +  1  =  0.    Setting  y-| —  =  e, 

we  get  z^  —  Az  -\-  d  =  0,  whence  2  =  1  or  3.     Hence 

y^  -.y^l  =  0,     or    y?  -  3y  +  1  =  0. 
.-.    y  =  id  ±  ^^3"),     i(3  ±  i/5  ),  or  1. 

EXERCISES. 

Solve  the  following  reciprocal  equations: 
1.  y6  ^  1  ^  0.  2.  y'  ^1  =  0.  3.  2/^  4-  1  =  0. 

4.  3^  _  5^2  __  5y  _|.  1  ^  0.  5.  y-^  -7y'-{-y'  -y'-i-7y-l  =  0. 

e.  y"  -  Sy'  {-Sy  -1  =  0.  7,  y'  -  lOy'  +  26^-^  -  lOy  i- 1  =  0. 

8.  2/10  _  3^8  ^  5^6  _  5^4  4-  32/2  -  1  =  0. 
Solve,  by  the  method  of  §  179,  Ex.  2,  the  equations 

2  1 

9.  y''  +  P2^^  -h  i^pV  -{~Y'P^y  +  f  =  o. 

10.  2/'  +  py'  +  |;?V  +  3^2  ^'2^'  +  ^  P^y  +  ^  =  0. 

4  7  5  1 

11.  2/"  +  py^  +  ^pV  +  jja^^^/'  +  jY^P'y^  +  f[iP^y  +  r  =  0. 

12.  y5  _!_  10^3  _^  20y  +  31  =  0. 

13.  y^  +  20f  4-  180y  -  211  =  0. 

181.  The  Fundamental  Theorem  of  Algebra.  An  equation  of 
the  nth  degree  has  exactly  n  roots. 

From  our  solution  of  the  general  equations  of  the  degrees  2,  3, 
and  4,  we  conclude  that  they  have  exactly  2,  3,  4  roots,  respectively. 
In  the  Appendix  it  is  proved  that  any  equation  has  a  root,  whatever 
be  the  degree  of  the  equation.     It  may  then  be  readily  shown  that 


194  THEORY  OF  EQUATIONS,  [Ch.  XIX 

an  equation  of  the  ni\\  degree  has  exactly  7i  roots.     Let  the  given 
equation  be 

(15)  f(x)  =  p^x-^p^x^-^J^p^x--'-{. .  . .  j^p^_^x-^2j^=^Q     (p^  ^  0). 

By  the  theorem  cited,  it  has  a  root,  say  a,  so  that  f{a)  =  0.     By 
the  factor  theorem,  f{x)  is  divisible  hj  x—  a: 

f{x)  =  {X  -  a)Q{x),      Q(x)  ^p,x—'  +  q,x--'+,  .  .  +  q,_,. 

Also  the  equation  Q(x)  =  0  has  a  root,  say  /?,  so  that 

Q{x)  =  {x-  fi)Q'{x),     Q\x)  ^p,x—'  +  rx—'  +,,. 

r.    f(x)  =  {x-a){x-^)Q'{x), 

so  that  /3  is  also  a  root  of  the  original  equation  f{x)  =  0.     Proceed- 
ing in  this  way,  we  get 

(16)  fix)  =p^{x-a){x  -  ^){x  -y),..{x^v). 

But  a  product  vanishes  if,  and  only  if,  one  of  its  factors  vanishes ; 
and  X  —  a,  for  example,  vanishes  if,  and  only  if,  x  =  a.    Hence  the  , 
equation  has  the  n  roots  a,  /3,  y,  .  .  . ,  rand  no  other  roots,  if^^^^O. 

In  case  a  —  p,  with  a  different  from  the  remaining  roots,  we 
say  that  the  equation  has  a  double  root  a.  In  case  a  —  ^  —  y^ 
with  a  different  from  the  remaining  roots,  the  equation  is  said  to 
have  a  triple  root.  In  general,  it  may  have  a  root  of  multiplicity 
2,  3,  4, .  .  . ,  or  n. 

As  shown  in  §§  70,  77,  an  equation  of  the  n\\\  degree  having 
more  than  n  roots  has  all  its  coefficients  equal  to  zero  and  is  an 
equation  only  in  appearance. 

If  follows  that  the  equation  whose  roots  are  a,  /?,...,  r  is 

(x  -  a){x'-  ^)..,  (x  -  v)  =  0. 

For  example,  the  equation  having  the  roots  1,  2,  —  1,  —  2  is 

(x  -  l){x  -  2){x  +  l){x+2)~  (x^-l)  {x^-  4)  =  x'-  5a;2  +  4  r=:  0. 

182.  Relations  between  the  roots  and  the  coefficients  of  an 
eq^uation.     As  illustrations  of  the  general  theorem  now  to  be  proved. 


I 


Sec.  183]  COLLEGE  ALGEBRA,  I95 

the  case  of  the  quadratic  equation  (§  32),  the  case  of  the  cubic 
equation  (§  174  and  Exercise  7),  and  the  case  of  the  quartic  equa- 
tion (§176)  should  be  reviewed  by  the  student. 

Suppose  that  in  an  equation  (15)  of  the  nth.  degree  the  coefficient 
Po  of  the  highest  power  x"'  is  unity. ^  Comparing  it  with  (16),  we 
have 

a;"-fi?,i?;"-i+;?2^^'"'+-  •  •  +;?^-l^+i^=(^-«')  (^-/^)(^-7)  •  •  •  {x-r). 

Upon  multiplying  together  the  n  factors  in  the  second  member  and 
denoting  by  8^  the  sum  of  the  roots  a,  /S,  .  ,  ,  ,  ix,  v,  by  S^  the  sum 
of  their  products  taken  two  at  a  time,  etc.,  as  written  below,  we  get 

X-- s,x^-^+8,x^-^-s^x^-^  + ...  ^  {-iY-\s^_^x -\-  {-lys^. 

Since  this  expression  must  be  identical  with  x"^  -\- ,  ,  .  -\-  Pny  we  get 

p\'=  S,^a^+ay  +  /3y  +  ,..-\-yv  +  fxv, 
(17)  \     -p,=-S,  =  a/3y+a^M  +  ''^  +  rM^. 


If  the  coefficient  of  x"^  is  unity,  the  7iegative  of  the  coefficient 
of  x""-^  equals  the  sum  of  the  n  roots;  the  coefficient  of  x''~'^  equals 
the  sum  of  the  products  of  the  roots  taken  tiuo  at  a  tijjiej  the  negative 
of  the  coefficient  of  x^~^  equals  the  sum  of  the  products  of  the  roots 
tahen  three  at  a  tiine,  etc;  finally,  {~^Y  H'^^^^s  i^f^^  constant  term 
equals  the  product  of  the  roots. 

Corollary.  If  the  roots  of  an  equation  are  all  positive,  the  coef- 
ficients 1,  ^1,  p^,  p^y^ .  .  .  Pn  are  alternately  positive  and  negative. 
If  the  roots  are  all  negative,  the  coefficients  are  all  positive. 

183.  Since  we  have  n  relations  (17)  for  the  n  roots,  it  might  be 
supposed  that  these  relations  would  enable  us  to  determine  the  roots. 

*  If  it  is  not  unity,  we  divide  the  equation  by  p^  and  use  the  new  equation. 


196  THEORY  OF  EQUATIONS.  [Ch.  XIX 

To  show  that  the  determination  of  one  root  a  by  means  of  these  n 
relations  requires  the  sohition  of  the  given  equation  of  the  nth. 
degree,  we  consider  the  case  n  =  3,  the  method  being  general.   Then 

Multiply  the  first  relation  by  a^^  the  second  by  —  a,  and  adding  to 
the  third  relation,  we  get  —p^  a^  —p^a  —  p^  =  a^, 

. •,      ^3  _|.  ^y^  ^2  j^  p^a-}-p^  =  0, 

which  is  the  given  cubic  x^  +  i^i  ^^  +  /'a  ^  +  /'a  =  0  with  a  written 
in  the  place  of  x.  Evidently  any  other  method  of  eliminating  /3 
and  y  from  the  three  relations  must  lead  to  the  same  result. 

But  the  relations  between  the  roots  and  the  coefficients  are 
nevertheless  of  both  theoretical  and  practical  value  in  problems  on 
the  solution  of  the  equation. 

184.  Example  1.     Solve  the  equation  a^  -]- 6x^  —  16x  —  80  =  0,  given  that  , 
one  root  is  the  negative  of  another  root. 

Since  the  roots  are  a,  /3  =  —  a,  y,  we  have 

—  ^=  a-\-  fi-\-y  =zy,     -  16  =z  afJ -\- ay -\- fty  =  -  a\    SO  =  afiy  =  - a^. 

Hence  y  =  —  6,  a=±4:,  ^=t4:.     The  roots  are  thus  —  5,  4,  —  4. 

Example  2.  Solve  the  equation  ar^  —  t6x^  +  7ix  —  105  =  0,  being  given 
that  the  roots  are  in  arithmetical  progression. 

Denote  the  roots  by  c  —  d,  c,  c-\-d.  Their  sum  3c  eqaals  15,  whence 
c  =  5.     Also 

(c  -  d)c  +(c-  d){c  +  c7)  +  c(c  +  (?)  =  3c2  -  (?  2  ^  71. 
Hence  d'^-4:,d=±2.     The  roots  are  thus  3,  5,  7. 

Example  3.  Solve  the  equation  x^  +  20x^  —  21005^  —  540a;  +  729  =  0,  the 
roots  being  in  geometrical  progression. 

Denoting  the  roots  by  a,  ar,  ar\  ar^,  we  get 

-  20  =  a(l  4-  r  +  7-2  +  r^)  =  a{i  +  r)(l  +  r% 
-  210  =  a\r  -j-r^  -{-2r^-\-r*  +  r^)  =  «V(1  +  r  +  r'')(l  +  r^), 
540  =  a^(7^  4-  7^  -f  7-5  +  7^)  =  a^r\l  +  r)(l  -f  r'), 
729  =  ^47^. 


I 

i 

Sec.  184]  COLLEGE  ALGEBRA,  197 

By  the  first  and  third  relations,  a^r^  =  —  27,  and  the  fourth  relation  is  satis- 
.fied.     Substituting  the  value  of  a^  in  the  second  relation,  it  becomes 


210  =  27(l  +  r  +  i)(r+i). 


which  is  evidently  a  reciprocal  equation  (§  180). 

,17  -10 

•••     ^  +  7=8'     "^    -3- 

Using  the  second  value,  we  get  r  =  —  3  or  —  i.     For  r  =  —  3,  we  get  a^  =  1 ; 
but  the  first  relation  gives  a  =  -\-l.     The  roots  are  1,  —  3,  9,    —  27. 

Example  4.     Solve   the  equation  ^a^  —  llx^  -\-  ^x  —  1  —  ^,   being  given 
that  the  roots  are  in  harmonical  progression. 

••     c-d~^c~^c  +  d~^'       c{c-dy    c{c-^d)'^  c' -d''        * 

1  1 


c{c  —  d){c -]- d)       6 

By  the  last  two  relations,  (c  -|-  (Z  )  -f  (c  —  cZ )  +  c  —  6,  whence  c  =  2.  The  third 
relation  then  gives  (2  —  d){2  -\-  d)  =  3,  whence  d^  =  1.  The  roots  are  there- 
fore 1,  ^,  J. 

Example  5.     Given  that  one  root  is  double  another,  solve  the  equation 

24a;3  _|_  i4a,2  _  g3^  _  45  ^  0. 
The  roots  being  a,  ft  —  2a,  and  y,  we  have 

;[         g  25         5 

By  the  first  and  second  relations,  a  =  —  or  —^,  y  —  or  — .      But  the 

1  —  25 

pair  of  solutions  a=  --,  y  =  — ttt"  ^^  ^^^  satisfy  the  third  relation,  whereas- 

-^  ^   A         u  .1.  .  -3-35 

the  pair  a  —  --r-,;K  =  5-  do.     Hence  the  roots  are— 7-,  -— r-,  — . 

EXERCISES. 
Form  the  equation  whose  roots  are 
1.    -3,  +  3,  1.         2.    ±  1,  ±  2,  0.         Z.  ci-d,  c-  d,  c,  d. 
4.  1,  1  4-   1/3;  1  -   |/3.  6.  -  1,  2  +  3  V~^»  2^3  j/^^l. 


i. 


198  THEORY  OF  EQUATIONS,  [Ch.  XIX 

6.  Solve  4:0^  —  16a;2  —  9a;  +  36  =  0,  one  root  being  the  negative  of  another. 

7.  Solve  ar*  —  lOa?'*  +  27aj  —  18  =  0,  one  root  being  double  another. 

8.  Solve  a^  —  dx"^  -{-  2dx  —  15  =  0,  one  root  being  triple  another. 

9.  Solve  a;*  —  4a^  —  2x^  -\-  12x  -j-  9  =  0,  having  two  pairs  of  equal  roots. 

10.  Solve  x^  —  23?  —  21x^  -\-  22x  +  40  =  0,  whose  roots  are  in  arithmetical 
progression,  say  c  —  36,  c  —  5,  c  +  6,  c  +  3&. 

11.  Solve  x^  —  14a;2  —  84.^  +  216  =  0,  whose  roots  are  in  G.P. 

12.  Solve  2W  —  42x2  _  28a!  +  8  =  0,  whose  roots  are  in  G.P. 

13.  Solve  ^x^  -  40a;3  _|_  i^(^^2  _  ^^{^x  +  27  =  0,  whose  roots  are  in  G.P. 

14.  Solve  24a;^  —  26a;2  -f  9^  —  1  =  0,  whose  roots  are  in  H.P. 

15.  Solve  ^Ix^  -  \W  -  36aj  +  8  =  0,  whose  roots  are  in  H.P. 

16.  Solve  the  equation  in  Ex.  15,  one  root  being  the  negative  of  another. 

17.  Solve  the  equation  in  Ex.  14,  one  root  being  double  another. 

18.  If  the  roots  of  a;^  —  px^  -{-qx  —  r  =  0  are  in  G.P.,  then  q^  =  ph\ 

19.  If  one  root  of  x^  —px^  -\-qx  —  r  =  0  is  the  negative  of  another,  pq  =  r. 
20    Solve  a;*  —  6a^  +  12a;2  —  10a;  +  3  =  0,  having  a  triple  root. 

21.  Solve  x^  -  Idx*  +  67a;3  -  171a;2  +  216a;  -  108  =  0,  having  a  double  root 
and  a  triple  root. 

185.  Fractional  Roots.  An  equation  whose  coefficients  are 
integers,  that  of  the  first  term  being  unity,  has  no  fractional  root. 

For  if  the  fraction  -,  where  a  and  b  are  integers  having  no 

common  divisor  greater  than  unity,  is  a  root  of  the  equation 

then,  setting  ^  =  "t  and  multiplying  the  equation  by  Z>*'~S 
j  +  p^a--'+p,a^-'b  +  ...+Pn-iab-~'+pJ--'  =  0, 

SO  that  the  fraction  -^  would  be  expressible  as  an  integer,  which  is 

contrary  to  the  hypothesis  on  a  and  b, 

186.  Surd  Roots.  If  an  equation  with  rational  coefficients  has 
the  root  a  -j-  4/^,  ivhere  a  and  b  are  rational,  but  \/b  is  irra- 
tional, then  the  equation  has  the  root  a  —  \^b. 


Sec.  187]  COLLEGE  ALGEBRA.  199 

Let  a  +  \/h  be  a  root  of  f{x)  =  0,  an  equation  with  rational 
coefficients.     Consider  the  product 

(x  —  a  —  \/b){x  —  a-\-  \/h)  =  {x  —  aY  —  b^x^  —  2ax  -\-d^  —  h. 

Divide  this  quadratic  expression  whose  coefficients  are  rational  into 
the  function  f{x)  and  denote  the  quotient  by  Q  and  the  remainder  by 
Rx  +  R',  where  R  and  R'  are  rational  numbers  independent  of  x, 

.-.    f{x)^Q{{x-af -h\+Rx-\-R\ 

This  identity  holds  for  any  value  of  x.  Taking  x  =^  a  -\- 1/^,  the 
left  member  is  zero  by  hypothesis,  also  {x  —  a)'^  —  b  vanishes. 

I  .-.     0  =  R(a  +  i/b^)  +  R\ 

By  §  5,  this  is  possible  if,  and  only  if,. 

r  0  =  Ra  +  R',    0  =  R. 

Hence  R=0,  R'  —  0,  so  that  f{x)  is  divisible  by  {x—aY  +  ^  and 
therefore  by  the  factor  x  —  a  -{-  \/b  oi  the  latter.  Hence  a  —  \/b\^ 
a  root  oif{x)  —  0. 

187.  Imaginary  roots.     If  an  equation  loith  real  coefficients  has 
the  root  a  -\-  b  \^  —l^  tvhere  a  and  b  are  real,  it  has  also  the  root 


a  —  b  \/  —  1,  so  that  imaginary  roots  occur  in  pairs. 

Let  a  +  ib,  where  i  =  4/  —  1,  be  a  root  off{x)  =  0,  an  equation 
with  real  coefficients.     Consider  the  product 

{x  —  a  —  ib){x  —  a  +  ib)  =  {x  —  a)^  +  P. 

Divide /(a;)  by  {x  —  aY  +  b'^  and  let  the  quotient  be  Q  and  the  re- 
mainder be  Rx-\-R',  where  R  and  R'  are  real. 

.',    f{x)^Q{{x~af^b^\-^Rx+R\ 


200  THEORY  OF  EQUATIONS,  [Ch.  XIX 

For  X  =z  a  -\-  ib,  f(x)  =  0  by  hypothesis.     Hence 

0  =:  ^(a  +  ih)  +  R. 

By  §  9,  this  is  possible  if,  and  only  if, 

0  =  Ea  +  Bi,     Eb  =  0. 

Hence  E  =  0,  E'  =  0,  so  that/(2;)  is  divisible  by  {x  —  ay^  -\-  b^  and 
therefore  by  the  factor  x—a-\-ib.    Hence  a  —  ib  is  a  root  oif(x)  =  0. 

EXERCISES. 

1.  Solve  a^  —  10«2  -|-  27a;  —  18  —  0,  the  roots  being  rational. 

2.  Solve  x^  -  2x^  —  2\x^  +  23a;  +  40  =  0,  the  roots  being  rational. 

3.  Solve  x^  —  Ax^  -\-  4:X  —  1  =  0,  one  root  being  2  -\-  |/3. 

4.  Solve  x^  -  36i»2  _  72^  _  36  =  0,  one  root  being  -  3  +  |/3. 

5.  Solve  x*'  —  ix^  +  ^x'^  —  2a;  —  2  =  0,  one  root  being  1  —  ^~—  1. 

6.  Solve  ar^  —  Sa;^  —  6a;  —  20  =2  0,  one  root  being  —  1  +  |/  —  3. 

7.  Solve  a;^  -  (4  -f  V3  )a;2  +  (5  +  4  ^3  )a;  -  5  |/3  =  0,  one  root  being  • 

8.  If  4/3  is  a  root  of  the  equation  in  Ex.  7,  is  —  4/3  also  a  root? 

9.  A  root  of  x^-{i  +  d)x^  +  (42;  +  9)a'— 5i— 5=0  is  1+^*.     Is  1  —  i  a  root? 
10.  Solve  a;5  +  a;*  —  Sx'^  —  9a;  +  15  =  0,  with  the  roots  |/3  and— 1— 2  i/^^. 

188.  Symmetric  functions  of  the  roots.     A  function  of  the  roots 
^f  /^9  y?  '  '  '  f  M>  ^  of  ^^  equation 

(18)  a:«+i^:r--^+;7,.T"-^+.  .  .  +Pn-ix  +  pn  =  0 

is  said  to  be  symmetrical  (§  33)  if  all  the  roots  are  similarly  in- 
volved.    For  example,  the  functions 

«  +  /^  +  r  +  •  •  •  +  /^  +  ^^    ^/^  +  ^r  +  /^r  + ' '  '  +  M^, 

a/3y  +  .  .  .  +  y/^iv,  .  .  .  ,    a^y  .  .  ,  fAv 

are  symmetric  functions  of  the  roots,  and  their  values  are  ~  p^,  p,^,  * 
--/?3,  ...,(—  lypny   respectively  (§  182).      More  generally,  any 


I    ■       . 

Sec   189]  COLLEGE  ALGEBRA.  201 

rational  symmetric  function  of  the  roots  can  be  expressed  rationally 
in  terms  of  the  coefficients  of  the  equation.  We  shall  prove  certain 
interesting  cases  of  this  theorem. 

Example  1.  To  find  the  sum  of  the  squares  of  the  roots. 

=  {- p,f  -  2{p,)  =  p,'  -  2p,. 

Example  2.  To  find  a'fJ  +  a/J^  +  aV  +  a/-^  +  . .  .  -f  //V  -f  //r^  =  2a'/^. 
Multiplying  together  the  equations 

we  get  2a'^/3  +  Saf^x  +  3a/5yW  +  .  .  .  -f-  Syjuv  =  —  p^p^- 

.-.     :2a' 13  =  3^3  ~p^2' 
Example  3.  To  find  the  sum  of  the  cubes  of  the  roots. 

(a  +  y5  +  .  .  .  4-  /^  +  y)icc'  +  f3'  +  ...+M'  +  y') 

I  =  (a^  +  ^^  +  .  .  .  +  y')  +  Ui'/3  +  ay^^  +  . . .  +  /^v  +  jiiy') 

Applying  the  results  of  Examples  1  and  2,  we  get 

189.  Denote  by  2^,  2^,  ^3,  :  .  ,  ,  2^  the  sum  of  the  1st,  2nd, 
3rd,  .  .  .  ,  rth  powers,  respectively,  of  the  roots.  Then  2^=  —  p^^ 
2^  =  p^  —  ^\,  ^3  =  —  Pi  +  ^ViP2  —  ^Pz^  ^y  Examples  1  and  3 
of  the  last  section.     These  results  give  the  relations 

(19)  2,+p,=  0,     2^+p,  :S,+2p,=  0,     ^3+;.,  2^+p^  ^\+3a=  0. 

I        Another  relation  of  similar  form  is  derived  by  substituting  in 
turn  a,  /3,  y,  ,  .  .  ^  r  for  x  in  the  given*  equation  (18).     Then 

a-+p^a^-'+p,a--'  +  ,  .  .  +  Pn-i  a  +  p,=  0, 


202  THEORY  OF  EQUATIONS,  [Ch.  XIX 

Adding  the  resulting  n  relations,  we  get 

(20)  2,+p,2^-i+p,2n-2+*  .  '+Pn-i^,  +  np^  =  0. 
We  proceed  to  show  that,  if  r  is  any  positive  integer  ^  71, 

(21)  2^+p^2^._i+p^2^_^  +  .  .  .+p^_^2^+rp,,=  0    \r=ny 

Substitute  for  p^,  .  .  .  ,  p^.  their  expressions  (17)  in  terms  of  the 
roots.     Then  * 

—  (ay^^-i  +  of^-i/J-f  .  .  .  +^i/''-i-|-;i'— V), 

p^  :^V_2  =     (^^^-'  +  «^-^^  + . . .  +  M^'^-'  +  ^'-'y) 

-  {a/Syd''-^  +  a/Syv"-^  +...)» 
jt7,_,  :^,  =  (-  1)'-H«^r .  .  .  p'^  +  t./? .  .  .  pcT^  +  .  .  .) 

rpr  =  (-  lYr{a/3y ,  . .  pa  +  ,  ,  .), 

Upon  adding  the  second  members,  we  observe  that  each  quantity  in 
a  parenthesis  on  the  extreme  right  cancels  the  quantity  in  the 
parenthesis  on  the  left  in  the  line  below,  so  that  the  sum  is  zero. 

*  The  relations  follow  immediately,  except  perhaps  that  for  pr  _  i^^  which 
equals  the  product  of  (—lV~\cc/3x  . . .  7tp-\-apy . .  .  7ta-\-afiy , .  .  pcr-f-  . . .) 
and  2i^cc-\-/3-\-y-\-...-\-7t-{-p-{-a-{-.,.-\-v.  There  is  a  single 
way  to  reach  the  term  afty  .  .  .  Ttp"^  in  the  product,  but  r  ways  to  reach  the 
term  a/3y  .  .  ,  Ttpa,  viz.,  by  using  either  a,  fi,  y,  .  ,  ,  ,  7t,  p,  or  cr  from  the 
sum  ^1. 


Sec.  190]  COLLEGE  ALGEBRA.  203 

Suppose  next  that  r  >  n.  For  the  case  r  =  7i-\-l,  we  retain 
the  above  relations  with  the  exception  of  the  last  two.  The  one 
preceding  the  last  is  no  longer  true,  the  correct  result  being  now 

=  (—  lY{a/3y .  .  .  /uv-  +  a/3y .  .  .  yuV  -{-  .  .  .  -|-  a^/3y .  .  .  /uv). 


\ 


Upon  adding  the  latter  to  the  n  preceding  relations,  we  get 
In  a  similar  manner,  we  may  prove  the  formula 

{^^)^n  +  yn+Pi^n^m-l+P2^n^m-2+'   •   ' +Pn-l  ^m  +  1+Pn  ^m  =  0. 

But  the  last  formula  is  derived  more  simply  as  follows.  Multi- 
plying the  given  equation  (18)  by  x^,  we  get 

^^n4-m_j_^^^n  +  m-l_|_^^^n^m-2_|_^  _  ^  ^^  _^  ^m  +  1  ^  ^^  ^m  _  q^ 

Setting  x=a,  /3,  y,  ,  ,  ,  ,  /A^  V  in  turn  and  adding  the  7i  resulting 
relations,  we  evidently  obtain  formula  (*22). 

190.  Relations  (21),  which  include  relations  (9),  enable  us  to 
express  2^,  2^,  2^,  .  .  •  ,  ^n^^  terms  of  the  coefficients.  We  have 
only  to  take  r  =  1,  2,  3,  .  .  .  ,  ^^  in  turn,  the  respective  equations 
giving  JS'j,  ^25  ^s»  .  .  .  ,  ^n  in  turn.     For  example, 

^2=    -Pl^l-    ^P2   =   Pi      ~    ^P2^ 

^Z^    -Pi   ^2+ Pi  P2    -    3^3  =    -  Pl-\-^PlP2  -   ^PV 

^4  =    -   Vl   ^3   -'P2^2-  Pz  ^l    -   ^Pi 

=  p,*  -  4p^%  +  4^j  p^  +  2p,^  -  4p^. 

Then  by  employing  formula  (22)  for  m  =  1,  2,  3, ...  in  turn,  we 
may  express  -S'^^j,  2n  +  2  9  ^n^-zf  >  -  -  in  terms  of  the  coefficients. 


204 


THEORY  OF  EQUATIONS, 


[Ch.  XIX 


EXERCISES 

1.  If  a,  fiy  y  are  the  roots  of  a^  -f  j^^  x^  -\-  Pi^  -^  P%  —  ^,  find 

1^3,  ^,,  2,,  2,,     {a'fi^  +  aV^  +  /jy^),     (a  +  ft){a  +  ^)(y5  +  ;.). 

2.  If  «:,  /5,  ;^,  d  are  the  roots  of  x^  -{-  Pi^^  -\-  Pi  ^^  +  it's  ^  +  i?4  =  0,  find 
^3,  ^„  :S5,  :S6,     {a'Py  +  a^yj^  +  .  .  .  +  /i;^^^),     (a^^  +  ay^s  _^  _    ). 

3.  Find  the  sum  of  the  reciprocals  of  the  roots  of  the  equations  in  Ex- 


amples 1  and  2.     Find  the  sum  of  the  squares  of  their  reciprocals. 
4.  Show  that  for  the  general  equation  (18) 

n    :Si 


(«  -  fif  +  {cz  -  rf  -{-  [fi  -  rf  -\- . , ,  =  n:s,  -  2,^  = 


(^  -  m^  -  rYifi  -ry  +  ^ 


n      -^1    ^2 
^1    ^2    ^3 

^2        -^3        •^4 


6.  Solving  equations  (19)  and  (21)  by  determinants,  prove  that 

;?i     1      0      0 
Fl      -       ^  c 

^3     -- 


2^2  = 


^1 


1  I 

^iV 


^Ps 


Pl 

1 

0 

2P2 

Pl 

1 

3^3 

Pi 

Pl 

^\ 

1 

0 

^1 

^1 

2 

^'3 

^^2 

^1 

^. 


24p^: 


2p,    p,     1      0 
^Ps     Pi     Pl      1 

4i?4  ;>3  ;?2   ;?i 

' 

:^i   1     0 
^1  2,   2 

^^3       ^2       ^1 

2,    2,     2, 

0 
0 
3 

6.  Form  the  equations  whose  roots  are  (1)  the  squares,  (2)  the  cubes,  of  the 
roots  of  ar^  +  3«2  +  aj  —  7  =  0. 

7.  If  a,  ^,  y  are  the  roots  of  x^  -[-  p^x^  -\-  p^x  -{-  p^  —  0,  form  the  equa- 


tions whose  roots  are  (1)  a:^ 
2    2    2 


y^;  (2)   a-2,  ^-2,  -^-2.    (3)  ^^^  ^^^  ^^  . 


r  f^  ,r 

ay       /3 


8.  Form  the  equation  of  lowest  degree  with  rational  coefficients,  one  of 
whose  roots  is  (1)  |/2  +  ^5  ;  (2)|/2  +  ^3  ;  (3)  i^S-  |/^;  (4)  |^^+  V^^- 


:    Sec.  190] 


COLLEGE  ALGEBRA. 


205 


MISCELLANEOUS    EXERCISES. 

.  .  +  71*  ^  Jq  ^(^^  +  l)(2/i  +  l)(3n2  +  37i  -  1). 
.  .  -I-  7^5  =  tV^'(^  4-  Ifi^n'  +  2/1  -  1). 
1      .       ,  .   .    ..  1 


1*  +  2*-+  , 

l^+2^  +  . 


4.  The  arithmetical  mean  of  two  positive  quantities  is  greater  than  their 
geometrical  mean. 

5.  According  as  ^  =  5   or  a  ^  &,  a^  +  5^  =  2ah  or  >  2ab. 

6.  If  (tt  +  Z>  +  c/  =  3(^2  _|_  ^,2  _^  ^2)^  tiieji  ^  ^  5  ^  ^^ 

7.  If  (^1-1-^2+.  .  .-\-a„fz=  n{a^-{-a^-{- .  .  .+an'0,  then  a^  =  a^=z  ,  .  .  =  «„. 

8.  According  as  .t;  -f  ^  +  2  >  ,  =  ,  or  <  0,  a^  +  ^^  -}-  2;3  >  ,  =  ,  or  <  3a?^2. 

9.  If  X,  y,  z  are  real  and  not  all  equal,  then 

mxyz  <  {x^  y  -\-  zf  <^{a^  -i-  y^  -\-^). 
10.  Prove  by  mathematical  induction  that 


a  +  («  +  ^)  +  («  +  2cZ)  +  . 

11.  Show  that  the  roots  of  a;^ 
1.3568958  .  .  . 


12. 
13. 
14. 

15. 


16. 


,  ^  {a  -{-  n  -  \d)  = 

7aj  -h  7  =  0  are 
1.69  .  .  .  ,   -  3.048. 
-  1  =  0  are  1.24698. 


<-+"-T^4 


The  roots  of  x'^  -\-  x^  —  2^;  -  1  =  0  are  1.24698,   —  1  80194,  —  .44504. 
The  positive  root  of  Ix^  +  20;^^  -\- ^x^  -  16aj  —  8  =  0  is  0.91336. 


0 
•  a 

■  1) 

■  c 


a  b 

0  d 

-d  '  0 

-e  -f 


:{af-  be-\-  cd)\ 


A  +  ri 
A  +  r2 


r  ^-  (X 

Yx  +  «^i 


a  +/? 
^i  +  /A 
^2  +  P'l 


-  2 


17.  Factor  .t*  -  17^^  ^  20aj  -  6,     x^  +  8.c3  +  ^Ix"  +  26«  +  14. 

18.  a?*  +  «"^  —  2^2  -|-  4ic  -  24  =:  0  has  two  real  and  two  imaginary  roots. 

19.  Factor  {a  +  1)^  ~  d^  -  \. 

20.  Solve  the  reciprocal  equation  27a^^il  —  xf  —  4t{l  —  x  -\-  x'^f. 

21.  li  a,  Py  y  are  the  roots  of  ic^  +  a!^  —  2a;  —  1  =  0,  form  the  equation 
a^  +  r'>  P'  +  r'- 


whose  roots  are  a^  +  ^^, 
22 


Sum  to  n  terms  ttt  +  tt  4- 
3!      4! 


Sum  to  infinity  —  +  -^  -f- 


5! 
3^ 
33 


•     1^   4- 

"^  "67  +  ' 


+  « 


2o6  THEORY  OF  EQUATIONS.  [Ch.  XIX 

„^    2    ,   12  ,   28  ,   50  ,   78   ,   112  ,  ^     ,  ^  , 

^^•ri+2-I+3!+4!+5-l+^  +  --  =  ^^  +  ^- 
23       33       43 

2*-i'  +  r!  +  2!+31  +  ---=-^^^- 

26.  Find  the  number  of  shot  arranged  in  a  triangular  pyramid,  there  being 
25  shot  along  each  edge. 

27.  If  the  sides  of  a  right-angled  triangle  are  in  A.  P.,  they  are  proportional 
to  3,  4,  5. 

28.  At  tennis  A  and  B  play  together  with  the  respective  winning  proba- 
bilities a  and  h.  If  the  score  is  deuce,  show  that  A's,  probability  of  winning 
the  game  is 

a^  +  %d'h  +  4a*6^  +  ^a^h'  +  .  .  .  =  ,    ^  ^  ^. 


APPENDIX. 


Argand's  Diagram.  In  1806  Argand  gave  a  method  of  repre- 
senting complex  numbers  by  points  in  a  plane.  The  complex 
number  x  -\-  y  \/  —  1  is  represented  by  the  point  {x,  y)  whose  coor- 
dinates referred  to  two  fixed  perpendicular  lines  Ox  and  Oy  are  the 
real  numbers  x  and  y.     To  each  complex  number  x  +  iy  there  cor- 


^ 


Y 

P 

{x-^iy) 

/ 

y 

/e 

0 

X 

\ 

Fig.  14. 

responds  a  single  point  P,  and  to  every  point  of  the  plane  there 
corresponds  one  and  but  one  complex  number.  If  the  point  P 
moves  in  the  plane,  z^x  -\-  iy  changes  continually  in  value  and  is 
called   a  complex  variable.     The  length  of  the  line  OP,  viz.^ 

207 


2o8  APPENDIX. 


\/x^  +  y'^,  is  called  the  modulus  of  x  +  iy.  The  angle  6  =  XOP  is 
called  the  argument  oi  x  -\-  iy;  it  is  to  be  measured  counter-clock- 
wise from  OX, 

By  trigonometry,  the  sine  of  6  is  the  ratio  of  y  to  r,  the  cosine 
of  6*  is  the  ratio  of  x  to  r.     These  relations  are  written 

t/  X 

sin  6  =  —,    cos  0  =  —.    whence  sin^  6  +  cos'-^  6  =  1. 
r  r 

.'.     a;  +  zy  —  r(cos  6  -{-.i sin  6) . 

For  example,  \/2  +  i\/2  =  2(cos  45°  +  ism  45°), 

- 1  _^  1. 4/-T==cosl20°H-i  sin  120°,   i+i  |/3"*  =:  cos  60°+z  sin  60°, 

If  the  points  Pand  A  represent  x  +  iy  and  a^  -f  ia,  respectively, 
it  is  seen  that  the  point  which  represents  their  sum  {x^a)-\-i(y-\-a) 
is  given  by  completing  the  parallelogram  two  of  whose  sides  are  OP 
and  OA.  It  follows  that  the  modulus  of  the  sum  (the  length  of  the  ' 
diagonal)  of  two  complex  numbers  is  less  than  (in  a  special  case, 
equal  to)  the  sum  of  their  moduli. 

The  point  represepting  the  product  of  two  complex  numbers 
may  be  constructed  by  applying  the  following  theorem : 

T/ie  modulus  of  tM  product  of  two  complex  numbers  is  the  prod- 
uct of  their  moduli,  it^  argument  is  the  suin  of  their  arguments. 

Let  x-\-iy  =  r(cos /^-j- ^  sin  ^), 

x'-\-  iy'  =  r'(eos  6'  +  isin  6'). 
Then     {x  +  *'y)(^'  ~^  W')~  rr' {cos  6  -\-  i  sin  ^)(cos  6'  -\- 1  sin  6') 
=  rr' ;  (cos  B  cos  B'  —  sin  6  sin  &)  +  i  (sin  d  cos  6'  -\-  cos  6  sin  6')  \ 
r=.  r/-'jcos(/9  +  &)  +  i  sin  {B -\-  &)\, 
l-y  the  formulae  proved  in  trigonometry. 

Demoivre's  Theorem.     If  n  he  any  rational  number  and  6  any 
angle,  (cos  6  -\-  i  sin  BY  =^  cos  nB  -\-  i  sin  nB. 

We  have  just  shown  that 
(cos  B-{-i  sin  6')(cos  6'  +  i  sin  6')  =  cos  {B  +  B')  +  *  sin  ((9  +  B'}. 


COLLEGE  ALGEBRA.  209 

Taking  first  6'  —  d,  we  have 

{cos  6  +  i8m  8y=:  CO&26 +  ism20. 
Taking  next  6'  =  26/,  and  applying  the  last  result,  we  get 

(cos  0  -\-  i  sin  Oy  =  cos  36  -\-  i  sin  30. 
Taking  next  0'  =  30,  and  applying  the  previous  result,  we  get 

(cos  6  -^  i  sin  6y  =  cos  4:0  +  ^  sin  40, 

Similarly,  if  the  theorem  be  true  for  71  =  r,  a  positive  integer,  it  is 
true  for  n  ■=  r  -\-  l.  Hence,  by  mathematical  induction,  it  is  true 
for  every  positive  integral  value  of  ^. 

n 

Eeplacing  0  by  -,  we  get 

/      0  oy 

COS  — |-  I  sin  -     =  cos  0  +  {  sin  0, 
\       n  nl 

-  0  0 

.  • .  (cos  0  4-  i  sin  Oy  =  cos  — \-  i  sin  -, 

^  71  n 

as  one  of  the  fith.  roots  of  cos  0  -\-  i  sin  0.     It  follows  that  the 
theorem  is  true  for  positive  fractional  values  of  /^. 
Next,  it  71  =  —  m,  771  being  a  positive  fraction, 

(cos  0  -{-  i  sin  OY  =  7 ^   ,    .    . — 77- 

^  '  ^         (cos  0  4-  I  sm  ^)"* 

1  cos  7nO  —  i  sin  mO 


cos  mO  -{-  i  sin  7nO      cos^  mO  -|-  sin''^  mO 
=  cos  7n0  —  i  sin  mO  =  cos  7iO  -\-  i  sin  tiO. 

Corollary.   Any  root  of  a  complex  number  may  be  expressed  as  a 
complex  number.     Thus  ^x  -j-  iy  =  \/r  (cos  0  -\- i  sin^)n.     Hence 

— \-  i  sin  -1  is  one  of  the  7ii\i  roots  of  x  -\-  iy,     Eeplacing 

6/  by  6/  +  360°,  6>  +  2  •  360°,  ,  ,  , ,  0  ^  {71  -  1)  360°,  we  obtain  the 
remaining  w  —  1  roots.  But  0  -\-  71  360°  gives  the  same  root  as  0, 
e  +  {n  -\-  1)  360°  gives  the  same  root  as  (9  +  360°,  etc. 


2IO  APPENDIX. 

Example  1.     Find  the  three  cube  roots  of  unity. 

1  =  cos  0  + 1  sin  0  -=  cos  360°  +  i  sin  360°  =  cos  720o  +  i  sin  720^ 
Hence  the  cube  roots  of  unity  are 

cos  f  +  /  sin  I  =  1,     cos  120''  +  i  sin  120°  =  -  i  -f-  i  \/^  =  co, 
cos  240°  +  i  sin  240°  =  -  ^  -  J  \/'^  =  gd\ 
Example  2.  Find  the  five  fifth  roots  of  —  1. 
-1  =  cosl80°+^  sinl80°=  cos  540°+^sin540°=...=cos  1620°+ z  sin  1620^ 
,-.  i/Zl  =  cos   36°+isin36°,  or  cos  108°+^sinl08°,  ...,  or  cos  324° -|-^  sin  324°. 

Solution  of  the  cubic  y^  -\-  py  -\-  r  =  0  in  the  irreduciUe  case 
(see  §173): 

(1)  ^-T  +  ^<«-    . 

Set        i^  =  —  p^  sin^  6,     —  -  =  p  cos  6. 

.-.  p'^  =  p2  (cos^  e  +  sin'^  6)=z^  -E=  -£^, 


.  • .  p  z=  4/-  f/27,      cos  e  =  -  --^i/-  py27. 
Notice  that  p  must  be  negative  by  (1),  so  that  p  is  real;  also  that 


—  —  is  in  numerical  value  less  than  4/—  p^/27  by  (1),  so  that  tos  0 

is  numerically  less  than  unity.     Hence  6  may  be  found  by  means 
of  a  trigonometric  table  of  cosines. 


V 


—  —  4"  4/^  =  j/p  cos  6  -f-  ip  sin  6^ 


3       f         ^  _j_  M.360"     ,     .    .     ^  +  77/^360°) 
4/p  -^  cos +  I  sm  — ■ — l 


y/-^-i/B  =  Vp\ 


6+  m.360'       .    .    6'  +  m.3()0° 
cos z  sm 


3     r 

where  m  =:  0,  1,  or  2  and  Yp  ^^  ^  ^^^^  quantity.     Substituting  in 
Cardan's  formulae,  we  obtain  the  three  roots 

y  =  2|/p  cos  — ! — {m  =  0,  1,  and  2). 


COLLEGE  ALGEBRA.  211 

We  may   proceed   independently  of    Cardan's   formulae.       By 
trigonometry,  cos  3:^:  =  4  cos^  x  —  ^  cos  x.     Set  z  =  cos  x. 

.' .  z^  —  \z  —  \  cos  3.T  =  0. 
To  reduce  the  given  cubic  to  this  form,  set  y  =  kz. 

p        3  / 1_  4/p 

Determine  k  so  that  f^  =  ——,  viz.,  take  k  =  \/      ^  ^ .     The  two 
A;^  4  V       3 

equations  will  then  be  identical  if 

1  o  ^ 

—  -  cos  3a;  =  ^^ 

4  KT 


~8  "^  Y  "27^ 


Hence  cos  3^;  =  —  —  ^  4/— i?V27,  the  value  found  above  for  cos  6 

Finding  3:^;  from  its  cosine  by  means  of  a  table,  a  root  z  equals 
cos  X.  But  cos  ?>x  —  cos  {^x  +  360^)  =  cos  {Zx  +  720°).  Hence 
the  three  roots  z  are 

cos  X,     COS  (2;  +  120°),     cos  (x  +  240°). 
Multiplying  these  by  ^  =  2  4/—  /?/3,  we  get  the  three  roots  y, 

FUNDAMENTAL   THEOREM. 

Every  equation  z^  +  p^z''~'^  +  .  .  .  +  jf?„  =  0  has  a  root. 
If  we  set  w  =  2;**  +  j^ja;""""^  +  A^*""^  +  •  •  •  +  A  >  we  are  to  prove 
that  there  exists  at  least  one  complex  number  z  \fhich  will  make 
w  —  0,  so  that  the  point  (called  the  w-point)  in  Argand's  diagram 
which  represents  w  will  be  the  origin  0.  Unless  the  theorem  is 
true,  there  is  a  ^(^j-point,  say  P^,  corre^onding  to  some  value  z^  of 
z  and  distinct  from  0,  such  that  no  other  z^-point  is  nearer  to  0 
than  Pj ,  so  that  there  is  no  value  of  z  which  will  bring  the  corre^ 
sponding  lu-^omi  within  the  circle  about  0  of  radius  OPy  If  we 
prove  that  the  last  statement  is  false,  the  theorem  will  be  proved. 


212  APPENDIX, 

Let  us  change  z  from  z^  to  z^  +  ^i  by  adding  a  small  quantity 
Zy     Then  «^  will  change  from  w^  to  ]^^,  where 

w={z,-^z,Y+p,{z,+zy-^+p,{z,+z,Y-^~^,,.-^p^_,{z,+z,)+^^ 

upon  expanding  the  powers  by  the  binomial  theorem  and  arranging 
the  terms  according  to  powers  of  Z^     The  coefficient  of  Z^  is  seen  * 
to  be* 

A  =  nz;^-'  +  {n  -  l)p^z^--'^+  (n  -  2)p,z,^-'  +  ...  +Pn-i* 
Since  w^^z^+  VK~^  +  .  .  .  +  jt?^ ,  we  get 

W=w^'\-  AZ,  -f-  BZ^^  -^  ,  ,  ,  +  Z,\ 

Set  A  =  a  (cos  a  +  t  sin  a),   Z^—  r  (cos  ^  +  *  sin  ^).     As  above, 

AZ^  —  ar  [cos  (^  +  ^)  +  ^  sin  (^  +  o')]. 

.-.  W=  iv^+ar  [cos(^+«')  +  ^sin  {e-{-a)]-^C,r^-\-C^r^+,  .  .  +  (7„r% 
the  e^act  form  of       C^  ~  B  (cos  6^  +  ^'  sin  6y^,       G^^  .  .  . ,      (?„ ^ 
being  immaterial.     By  taking  r   sufficiently   small,   the  value   of 
C^r^  +  .  .  .  +  ^vT^  can  be  made  as  small  as  we  please,  and  there- 
fore W  will  differ  as  little  as  w^e  please  from 

w^  +  «^  [cos  {6  -\-  a)  -\-  i  sin  {6  -f-  a)']. 

But  the  point  representing  this  complex  quantity  will  describe  a 
circle  of  radius  ar  about  the  point  P^  when  the  angle  6  increases 
gradually  from  0°  to  360°.  ^ut  this  circle  intersects  the  circle 
about  0  with  radius  OP^  Hence  by  assigning  suitable  values  to 
r  and  0,  the  w-point  may  be  brought  into  coincidence  with  some 
PT-point  lying  inside  the  circle  of  radius  OP^ 

*  The  later  argument  assumes  that  A  4^  0.  Since  there  are  at  most  n  —  \ 
values  of  z^  which  make  ^  =  0  (§  77),  such  values  may  be  avoided  by  taking 
some  new  point  for  Pj  on  the  circle  of  radius  OP^, 


INDEX. 


(The  numbers  refer  to  pages.) 


Absolute  value,  1*14 

Annuity,  74 

Area  of  triangle,  156 

Argand's  Diagram,  207 

Argument,  208 

Arithmetical  progression,  64,  115 

Axes  of  coordinates;  153 

Bas  e  of  logarithms,  18,  20,  142 
Bino.raial  Theorem,  90,  101,  130 

Chance,   94 

Characteristic,  20 

Commensui-able,  58 

Common  diti*erenc(i,  64 

loga>ithms,  20 

ratio,  1^7  * 

Combination,  85 

Complex  quantity,  y5,  207  , 

Composition  and  divi^^ion,  61 

Compound  Interest,  73 

Conjugate,  7 

Constant,  76 

Convergent,  113 

Cubic  equation,  180,  210 

Cyclo-symmetry,  33 

Demoivre's  Theorem,  208 
Descartes'  Rule  of  Signs,  176 
Determinant,  second  order,  36 

third  order,  38 

fourth  order,  50 

Divergent,  114 

Double  root,  194 
Duplicate  ratio,  59 

Ellipse,  166 

Equating  coefficients,  78,  126 
Equation,  see  Cubic,  Quartic,  Fifth 
Exponential  equation,  26 


Exponential  Theorem,  137 
Extremes,  60 

Factor  Theorem,  27 
Fermat's  Theorem,  103 
Fifth-degree  equation,  189 
finite,  106,  113 
First  law  of  indices,  10„  13 
Fractional  root,  198 
Function,  76 
Fundamental  Theorem,  193, 


2ld 


Generating  fraction,  148 

relation,  145 

Geometrical  progression,  67,  114 
Graph,  154,  1^3,  168 

Harmonical  progression,  71,  116 
Horner's  method,  169 

Identity,  31  (Note),  78  (Note) 

Imaginary  quantity,  5 
' root,  199 

Incommensurable,  58,  144 

Independent  events,  96 

Indeterminate  form,  106 
InalV'es,  10 
Infinite,    105,  113 
Interest,  73 
Interpolation,  14}^ 
Irrational,  2,  3 
Irreducible  case,  184,  210 

Limit,  17,  69,  104,  110 
Location  of  roots,  178 
Logarithm,  18 

Mantissa,  20 

Mathematical  induction,  100 

Means,  60,  64,  67,  71 

213 


INDEX, 


Method  of  differences,  148 
Minors,  38,  50 
Miscellaneous  exercises,  205 
Modulus,  142,  208 
Multinomial  Theorem,  92 

Napierian,  142 
Natural  logarithm,  142 

number,  1 

Negative  number,  2 

Order  of  determinant,  .^6,  38, 

of  surd,  2 

Origin  of  coordinates,  153 

Parabola,  163,  168  '' 

Partial  fraction,  80 
Pascal's  Triangle.  88 
Permanence  of  form,  132 
Permutation,  85 
Perpetuity,  75 
Plot,  153 

Positive  integer,  1 
Power  series,  125 
Prime  number,  101 
Probability,  94 
Proportion,  60 

Quadratic  equation,  29 

surd,  2 

Quartic  equation,  185-188 

Ratio,  56,  67,  158 
Kational  number,  3 


j^     Sol 

X  Sqi 


/ 


$> 


^ 


Rational  integral  function,  77 
Rationalizing  factor,  29 
Real  number,  4 
Reciprocal  equation,  190 
Recurpng  series,  14^ 
Reduced  cubic,  181 
Remainder  Theorem,  27 
Reversion  of  series^  128 
Root,  30,  31,  194 


Second  law  of  indices,  10,  14  * 

■^  ries.  113 
limultaneous  equations,  35,  54,  164 

inequalities,  166 
Solution  of  numerical  equations,  168 
Square  root,  6,  8 
Sum  to  infinity,  69 
Surd,  2,  3 
Surd  root,  198 

Symmetrical  functions  of  roots,  200 
Symmetry,  33 
Synthetic  division,  169 

Table  of  logarithms,  24 
Third  law  of  indices,  11,  14 
Third  proportional,  60 
Triple  root,  194 
Triplicate  ratio,  59 

Undetermined  coefficie  nts,  78 

Vandermonde's  Thcjorem,  132 
Variable,  76 
Variation,  61 
Var}^  inversely,  61 


.^.^/■. 


^f  & 


'■^■^lik 


^ 


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